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logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Conditional Probability

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

Some stochastic experiments do not influence each other.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

Some stochastic experiments do not influence each other. For example, the probability that the second of two coin flips comes up heads is 1 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

Some stochastic experiments do not influence each other. For example, the probability that the second of two coin flips comes up heads is 1 2, no matter what the first coin flip was.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

Some stochastic experiments do not influence each other. For example, the probability that the second of two coin flips comes up heads is 1 2, no matter what the first coin flip was. (Yes, long “streaks” are unlikely, but the probability to win the next game does not change

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

Some stochastic experiments do not influence each other. For example, the probability that the second of two coin flips comes up heads is 1 2, no matter what the first coin flip was. (Yes, long “streaks” are unlikely, but the probability to win the next game does not change, even if the last 20 games were lost.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

Other stochastic experiments do influence each other.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

Other stochastic experiments do influence each other. For example, drawing cards from a deck without replacement, the probability that the first card is a queen is 4 52 = 1 13.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

Other stochastic experiments do influence each other. For example, drawing cards from a deck without replacement, the probability that the first card is a queen is 4 52 = 1 13. If a queen was drawn on the first try, the probability that the second card is a queen, too, is 3 51 = 1 17.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

Other stochastic experiments do influence each other. For example, drawing cards from a deck without replacement, the probability that the first card is a queen is 4 52 = 1 13. If a queen was drawn on the first try, the probability that the second card is a queen, too, is 3 51 = 1

• 17. In contrast, if the first

card drawn was not a queen, the probability that the second card is a queen is 4 51.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

So the probability of what looks like the same event can change depending on what happened before.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

So the probability of what looks like the same event can change depending on what happened before. As another example

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

So the probability of what looks like the same event can change depending on what happened before. As another example, (and to illustrate the difference)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

So the probability of what looks like the same event can change depending on what happened before. As another example, (and to illustrate the difference) the probability that you will run over a nail with your car’s tire does not change, no matter how long you drive on the tire.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Introduction

So the probability of what looks like the same event can change depending on what happened before. As another example, (and to illustrate the difference) the probability that you will run over a nail with your car’s tire does not change, no matter how long you drive on the tire. The probability that the nail will puncture your tire does, though.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

A

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

A B

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

A B

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

A B

• A∩B

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

A B

• A∩B

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

A B

• A∩B

B

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

A B

• A∩B

B

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

A B

• A∩B

B

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

A B

• A∩B

B

• A∩B

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Definition. For events A and B with P(B) > 0, the conditional

probability of A given that B has occurred is P(A|B) := P(A∩B) P(B) . S

A B

• A∩B

B

• A∩B

For P(A|B), the set B becomes the sample space.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen”

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

= 1 17

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

= 1 17 P(B′) = 48 52 = 12 13

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

= 1 17 P(B′) = 48 52 = 12 13, P(A∩B′)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

= 1 17 P(B′) = 48 52 = 12 13, P(A∩B′) = 48·4 52·51

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

= 1 17 P(B′) = 48 52 = 12 13, P(A∩B′) = 48·4 52·51 = 16 13·17

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

= 1 17 P(B′) = 48 52 = 12 13, P(A∩B′) = 48·4 52·51 = 16 13·17 P(A|B′)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

= 1 17 P(B′) = 48 52 = 12 13, P(A∩B′) = 48·4 52·51 = 16 13·17 P(A|B′) = P(A∩B′) P(B′)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

= 1 17 P(B′) = 48 52 = 12 13, P(A∩B′) = 48·4 52·51 = 16 13·17 P(A|B′) = P(A∩B′) P(B′) =

16 13·17 12 13

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

= 1 17 P(B′) = 48 52 = 12 13, P(A∩B′) = 48·4 52·51 = 16 13·17 P(A|B′) = P(A∩B′) P(B′) =

16 13·17 12 13

= 4 51

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

= 1 17 P(B′) = 48 52 = 12 13, P(A∩B′) = 48·4 52·51 = 16 13·17 P(A|B′) = P(A∩B′) P(B′) =

16 13·17 12 13

= 4 51 Proposition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

= 1 17 P(B′) = 48 52 = 12 13, P(A∩B′) = 48·4 52·51 = 16 13·17 P(A|B′) = P(A∩B′) P(B′) =

16 13·17 12 13

= 4 51

• Proposition. Multiplication rule: P(A∩B) = P(A|B)·P(B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A = “the second card drawn from a deck is a queen”

B = “the first card drawn from a deck is a queen” P(B) = 4 52 = 1 13, P(A∩B) = 4·3 52·51 = 1 13·17 P(A|B) = P(A∩B) P(B) =

1 13·17 1 13

= 1 17 P(B′) = 48 52 = 12 13, P(A∩B′) = 48·4 52·51 = 16 13·17 P(A|B′) = P(A∩B′) P(B′) =

16 13·17 12 13

= 4 51

• Proposition. Multiplication rule: P(A∩B) = P(A|B)·P(B)

(Can be used to double check and to compute probabilities of intersections.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Three cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all three cards must be turned over to find the queen of hearts?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Three cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all three cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Three cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all three cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Three cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all three cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. The event we are interested in is A1 ∩A2.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Three cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all three cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. The event we are interested in is A1 ∩A2. P(A1 ∩A2)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Three cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all three cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. The event we are interested in is A1 ∩A2. P(A1 ∩A2) = P(A1)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Three cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all three cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. The event we are interested in is A1 ∩A2. P(A1 ∩A2) = P(A1)P(A2|A1)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Three cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all three cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. The event we are interested in is A1 ∩A2. P(A1 ∩A2) = P(A1)P(A2|A1) = 2 3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Three cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all three cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. The event we are interested in is A1 ∩A2. P(A1 ∩A2) = P(A1)P(A2|A1) = 2 3 · 1 2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Three cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all three cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. The event we are interested in is A1 ∩A2. P(A1 ∩A2) = P(A1)P(A2|A1) = 2 3 · 1 2 = 1 3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. A3 := third card not the queen of hearts.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. A3 := third card not the queen of hearts. The event we are interested in is A1 ∩A2 ∩A3.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. A3 := third card not the queen of hearts. The event we are interested in is A1 ∩A2 ∩A3. P(A1 ∩A2 ∩A3)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. A3 := third card not the queen of hearts. The event we are interested in is A1 ∩A2 ∩A3. P(A1 ∩A2 ∩A3) = P(A3|A1 ∩A2)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. A3 := third card not the queen of hearts. The event we are interested in is A1 ∩A2 ∩A3. P(A1 ∩A2 ∩A3) = P(A3|A1 ∩A2)P(A1 ∩A2)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. A3 := third card not the queen of hearts. The event we are interested in is A1 ∩A2 ∩A3. P(A1 ∩A2 ∩A3) = P(A3|A1 ∩A2)P(A1 ∩A2) = P(A3|A1 ∩A2)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. A3 := third card not the queen of hearts. The event we are interested in is A1 ∩A2 ∩A3. P(A1 ∩A2 ∩A3) = P(A3|A1 ∩A2)P(A1 ∩A2) = P(A3|A1 ∩A2)P(A2|A1)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. A3 := third card not the queen of hearts. The event we are interested in is A1 ∩A2 ∩A3. P(A1 ∩A2 ∩A3) = P(A3|A1 ∩A2)P(A1 ∩A2) = P(A3|A1 ∩A2)P(A2|A1)P(A1)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. A3 := third card not the queen of hearts. The event we are interested in is A1 ∩A2 ∩A3. P(A1 ∩A2 ∩A3) = P(A3|A1 ∩A2)P(A1 ∩A2) = P(A3|A1 ∩A2)P(A2|A1)P(A1) = 1 2 · 2 3 · 3 4

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. Four cards are placed face down on a table. One of

them is the queen of hearts. What is the probability that all four cards must be turned over to find the queen of hearts? A1 := first card not the queen of hearts. A2 := second card not the queen of hearts. A3 := third card not the queen of hearts. The event we are interested in is A1 ∩A2 ∩A3. P(A1 ∩A2 ∩A3) = P(A3|A1 ∩A2)P(A1 ∩A2) = P(A3|A1 ∩A2)P(A2|A1)P(A1) = 1 2 · 2 3 · 3 4 = 1 4

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25, respectively.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15, respectively.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models? Name the models 1, 2 and 3, respectively.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models? Name the models 1, 2 and 3, respectively. B = purchased previously owned model again.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models? Name the models 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models? Name the models 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models? Name the models 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models? Name the models 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models? Name the models 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models? Name the models 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models? Name the models 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models? Name the models 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models? Name the models 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. A car company sells three different models of

compact cars. The probabilities that one of these models is selected (by a compact car buyer) are 0.40, 0.35 and 0.25,

• respectively. The probabilities that a buyer of any of these cars

will buy the same model again are 0.30, 0.05 and 0.15,

• respectively. Given that a buyer has just purchased a compact

car of the same model that was previously owned, what are the chances that the model was the first, second or third of the above models? Name the models 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.70

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.70 ✭✭✭✭✭✭✭✭ ✭s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.70 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.05

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.70 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.05 ❤❤❤❤❤❤❤❤ ❤s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.70 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.05 ❤❤❤❤❤❤❤❤ ❤s P(B′|A2) = 0.95

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.70 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.05 ❤❤❤❤❤❤❤❤ ❤s P(B′|A2) = 0.95 ✭✭✭✭✭✭✭✭ ✭s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.70 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.05 ❤❤❤❤❤❤❤❤ ❤s P(B′|A2) = 0.95 ✭✭✭✭✭✭✭✭ ✭s P(B|A3) = 0.15

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.70 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.05 ❤❤❤❤❤❤❤❤ ❤s P(B′|A2) = 0.95 ✭✭✭✭✭✭✭✭ ✭s P(B|A3) = 0.15 ❤❤❤❤❤❤❤❤ ❤s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.70 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.05 ❤❤❤❤❤❤❤❤ ❤s P(B′|A2) = 0.95 ✭✭✭✭✭✭✭✭ ✭s P(B|A3) = 0.15 ❤❤❤❤❤❤❤❤ ❤s P(B′|A3) = 0.85

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.70 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.05 ❤❤❤❤❤❤❤❤ ❤s P(B′|A2) = 0.95 ✭✭✭✭✭✭✭✭ ✭s P(B|A3) = 0.15 ❤❤❤❤❤❤❤❤ ❤s P(B′|A3) = 0.85 P(B∩A1) = P(B|A1)P(A1) = 0.30·0.40

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.70 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.05 ❤❤❤❤❤❤❤❤ ❤s P(B′|A2) = 0.95 ✭✭✭✭✭✭✭✭ ✭s P(B|A3) = 0.15 ❤❤❤❤❤❤❤❤ ❤s P(B′|A3) = 0.85 P(B∩A1) = P(B|A1)P(A1) = 0.30·0.40 P(B∩A2) = P(B|A2)P(A2) = 0.05·0.35

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.40 s P(A2) = 0.35 ❍❍❍❍❍❍❍❍ ❍s P(A3) = 0.25 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.30 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.70 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.05 ❤❤❤❤❤❤❤❤ ❤s P(B′|A2) = 0.95 ✭✭✭✭✭✭✭✭ ✭s P(B|A3) = 0.15 ❤❤❤❤❤❤❤❤ ❤s P(B′|A3) = 0.85 P(B∩A1) = P(B|A1)P(A1) = 0.30·0.40 P(B∩A2) = P(B|A2)P(A2) = 0.05·0.35 P(B∩A3) = P(B|A3)P(A3) = 0.15·0.25

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = P(A1 ∩B) P(B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = P(A1 ∩B) P(B) = P(B|A1)P(A1) P(B∩A1)+P(B∩A2)+P(B∩A3)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = P(A1 ∩B) P(B) = P(B|A1)P(A1) P(B∩A1)+P(B∩A2)+P(B∩A3) = P(B|A1)P(A1) P(B|A1)P(A1)+P(B|A2)P(A2)+P(B|A3)P(A3)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = P(A1 ∩B) P(B) = P(B|A1)P(A1) P(B∩A1)+P(B∩A2)+P(B∩A3) = P(B|A1)P(A1) P(B|A1)P(A1)+P(B|A2)P(A2)+P(B|A3)P(A3) = 0.30·0.40 0.30·0.40+0.05·0.35+0.15·0.25

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = P(A1 ∩B) P(B) = P(B|A1)P(A1) P(B∩A1)+P(B∩A2)+P(B∩A3) = P(B|A1)P(A1) P(B|A1)P(A1)+P(B|A2)P(A2)+P(B|A3)P(A3) = 0.30·0.40 0.30·0.40+0.05·0.35+0.15·0.25 = 0.12 0.175

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = P(A1 ∩B) P(B) = P(B|A1)P(A1) P(B∩A1)+P(B∩A2)+P(B∩A3) = P(B|A1)P(A1) P(B|A1)P(A1)+P(B|A2)P(A2)+P(B|A3)P(A3) = 0.30·0.40 0.30·0.40+0.05·0.35+0.15·0.25 = 0.12 0.175 ≈ 0.6857

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = P(A1 ∩B) P(B) = P(B|A1)P(A1) P(B∩A1)+P(B∩A2)+P(B∩A3) = P(B|A1)P(A1) P(B|A1)P(A1)+P(B|A2)P(A2)+P(B|A3)P(A3) = 0.30·0.40 0.30·0.40+0.05·0.35+0.15·0.25 = 0.12 0.175 ≈ 0.6857 ≈ 69%

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69%

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69% P(A2|B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69% P(A2|B) = P(A2 ∩B) P(B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69% P(A2|B) = P(A2 ∩B) P(B) = P(B|A2)P(A2) P(B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69% P(A2|B) = P(A2 ∩B) P(B) = P(B|A2)P(A2) P(B) = 0.05·0.35 0.175

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69% P(A2|B) = P(A2 ∩B) P(B) = P(B|A2)P(A2) P(B) = 0.05·0.35 0.175 = 0.1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69% P(A2|B) = P(A2 ∩B) P(B) = P(B|A2)P(A2) P(B) = 0.05·0.35 0.175 = 0.1 = 10%

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69% P(A2|B) = P(A2 ∩B) P(B) = P(B|A2)P(A2) P(B) = 0.05·0.35 0.175 = 0.1 = 10% P(A3|B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69% P(A2|B) = P(A2 ∩B) P(B) = P(B|A2)P(A2) P(B) = 0.05·0.35 0.175 = 0.1 = 10% P(A3|B) = P(A3 ∩B) P(B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69% P(A2|B) = P(A2 ∩B) P(B) = P(B|A2)P(A2) P(B) = 0.05·0.35 0.175 = 0.1 = 10% P(A3|B) = P(A3 ∩B) P(B) = P(B|A3)P(A3) P(B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69% P(A2|B) = P(A2 ∩B) P(B) = P(B|A2)P(A2) P(B) = 0.05·0.35 0.175 = 0.1 = 10% P(A3|B) = P(A3 ∩B) P(B) = P(B|A3)P(A3) P(B) = 0.15·0.25 0.175

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69% P(A2|B) = P(A2 ∩B) P(B) = P(B|A2)P(A2) P(B) = 0.05·0.35 0.175 = 0.1 = 10% P(A3|B) = P(A3 ∩B) P(B) = P(B|A3)P(A3) P(B) = 0.15·0.25 0.175 = 0.2143

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Models are 1, 2 and 3, respectively. B = purchased previously owned model again. P(B) = 0.175 A1 = purchased model 1. P(A1) = 0.40, P(B|A1) = 0.30. A2 = purchased model 2. P(A2) = 0.35, P(B|A2) = 0.05. A3 = purchased model 3. P(A3) = 0.25, P(B|A3) = 0.15. What is P(Ai|B)? P(A1|B) = 0.12 0.175 ≈ 0.6857 ≈ 69% P(A2|B) = P(A2 ∩B) P(B) = P(B|A2)P(A2) P(B) = 0.05·0.35 0.175 = 0.1 = 10% P(A3|B) = P(A3 ∩B) P(B) = P(B|A3)P(A3) P(B) = 0.15·0.25 0.175 = 0.2143 ≈ 21%

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Theorem. Law of total probability.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Theorem. Law of total probability. If A1,...,Ak are mutually

exclusive and exhaustive events

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Theorem. Law of total probability. If A1,...,Ak are mutually

exclusive and exhaustive events, then for any event B we have

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Theorem. Law of total probability. If A1,...,Ak are mutually

exclusive and exhaustive events, then for any event B we have P(B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Theorem. Law of total probability. If A1,...,Ak are mutually

exclusive and exhaustive events, then for any event B we have P(B)

• =

k

∑

i=1

P(B∩Ai)

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Theorem. Law of total probability. If A1,...,Ak are mutually

exclusive and exhaustive events, then for any event B we have P(B)

• =

k

∑

i=1

P(B∩Ai)

• =

k

∑

i=1

P(B|Ai)P(Ai).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Theorem. Law of total probability. If A1,...,Ak are mutually

exclusive and exhaustive events, then for any event B we have P(B)

• =

k

∑

i=1

P(B∩Ai)

• =

k

∑

i=1

P(B|Ai)P(Ai). Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Theorem. Law of total probability. If A1,...,Ak are mutually

exclusive and exhaustive events, then for any event B we have P(B)

• =

k

∑

i=1

P(B∩Ai)

• =

k

∑

i=1

P(B|Ai)P(Ai).

• Theorem. Bayes’ Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Theorem. Law of total probability. If A1,...,Ak are mutually

exclusive and exhaustive events, then for any event B we have P(B)

• =

k

∑

i=1

P(B∩Ai)

• =

k

∑

i=1

P(B|Ai)P(Ai).

• Theorem. Bayes’ Theorem. If A1,...,Ak are mutually

exclusive, exhaustive events with positive prior probabilities P(Aj), then for any event B with P(B) > 0 and any j, we have that the posterior probability of Aj given that B has occurred is

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Theorem. Law of total probability. If A1,...,Ak are mutually

exclusive and exhaustive events, then for any event B we have P(B)

• =

k

∑

i=1

P(B∩Ai)

• =

k

∑

i=1

P(B|Ai)P(Ai).

• Theorem. Bayes’ Theorem. If A1,...,Ak are mutually

exclusive, exhaustive events with positive prior probabilities P(Aj), then for any event B with P(B) > 0 and any j, we have that the posterior probability of Aj given that B has occurred is P(Aj|B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Theorem. Law of total probability. If A1,...,Ak are mutually

exclusive and exhaustive events, then for any event B we have P(B)

• =

k

∑

i=1

P(B∩Ai)

• =

k

∑

i=1

P(B|Ai)P(Ai).

• Theorem. Bayes’ Theorem. If A1,...,Ak are mutually

exclusive, exhaustive events with positive prior probabilities P(Aj), then for any event B with P(B) > 0 and any j, we have that the posterior probability of Aj given that B has occurred is P(Aj|B)

• = P(Aj ∩B)

P(B)

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Theorem. Law of total probability. If A1,...,Ak are mutually

exclusive and exhaustive events, then for any event B we have P(B)

• =

k

∑

i=1

P(B∩Ai)

• =

k

∑

i=1

P(B|Ai)P(Ai).

• Theorem. Bayes’ Theorem. If A1,...,Ak are mutually

exclusive, exhaustive events with positive prior probabilities P(Aj), then for any event B with P(B) > 0 and any j, we have that the posterior probability of Aj given that B has occurred is P(Aj|B)

• = P(Aj ∩B)

P(B)

• =

P(B|Aj)P(Aj) ∑k

i=1 P(B|Ai)P(Ai).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

Example.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs. P(A1) = 0.0001.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01. P(A1|B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01. P(A1|B) = P(A1 ∩B) P(B)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01. P(A1|B) = P(A1 ∩B) P(B) = P(B|A1)P(A1) P(B|A1)P(A1)+P(B|A2)P(A2)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01. P(A1|B) = P(A1 ∩B) P(B) = P(B|A1)P(A1) P(B|A1)P(A1)+P(B|A2)P(A2) = 0.98·0.0001 0.98·0.0001+0.01·0.9999

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01. P(A1|B) = P(A1 ∩B) P(B) = P(B|A1)P(A1) P(B|A1)P(A1)+P(B|A2)P(A2) = 0.98·0.0001 0.98·0.0001+0.01·0.9999 ≈ 0.0097

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

• Example. (Numbers made up.) A certain drug-sniffing dog will

alert its handler in 98% of all cases in which an individual is actually carrying drugs, but it will also alert its handler to 1%

• f the cases in which the individual is not carrying drugs. Let

us assume, as an estimate, that 1 in 10000 individuals checked actually is carrying drugs. The dog has just alerted its handler. Compute the probability that the individual is carrying drugs. A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01. P(A1|B) = P(A1 ∩B) P(B) = P(B|A1)P(A1) P(B|A1)P(A1)+P(B|A2)P(A2) = 0.98·0.0001 0.98·0.0001+0.01·0.9999 ≈ 0.0097 (about 1%)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001 ❍❍❍❍❍❍❍❍ ❍s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001 ❍❍❍❍❍❍❍❍ ❍s P(A2) = 0.9999

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001 ❍❍❍❍❍❍❍❍ ❍s P(A2) = 0.9999 ✭✭✭✭✭✭✭✭ ✭s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001 ❍❍❍❍❍❍❍❍ ❍s P(A2) = 0.9999 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.98

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001 ❍❍❍❍❍❍❍❍ ❍s P(A2) = 0.9999 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.98 ❤❤❤❤❤❤❤❤ ❤s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001 ❍❍❍❍❍❍❍❍ ❍s P(A2) = 0.9999 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.98 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.02

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001 ❍❍❍❍❍❍❍❍ ❍s P(A2) = 0.9999 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.98 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.02 ✭✭✭✭✭✭✭✭ ✭s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001 ❍❍❍❍❍❍❍❍ ❍s P(A2) = 0.9999 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.98 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.02 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.01

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001 ❍❍❍❍❍❍❍❍ ❍s P(A2) = 0.9999 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.98 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.02 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.01 ❤❤❤❤❤❤❤❤ ❤s

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001 ❍❍❍❍❍❍❍❍ ❍s P(A2) = 0.9999 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.98 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.02 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.01 ❤❤❤❤❤❤❤❤ ❤s P(B′|A2) = 0.99

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001 ❍❍❍❍❍❍❍❍ ❍s P(A2) = 0.9999 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.98 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.02 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.01 ❤❤❤❤❤❤❤❤ ❤s P(B′|A2) = 0.99

Fewer false positives would significantly increase the accuracy.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability

logo1 Events Influencing Each Other Conditional Probability Bayes’ Theorem

A1 = individual carries drugs. P(A1) = 0.0001. A2 = individual does not carry drugs. P(A2) = 0.9999. B = dog alerts handler. P(B|A1) = 0.98, P(B|A2) = 0.01.

s ✟✟✟✟✟✟✟✟ ✟s P(A1) = 0.0001 ❍❍❍❍❍❍❍❍ ❍s P(A2) = 0.9999 ✭✭✭✭✭✭✭✭ ✭s P(B|A1) = 0.98 ❤❤❤❤❤❤❤❤ ❤s P(B′|A1) = 0.02 ✭✭✭✭✭✭✭✭ ✭s P(B|A2) = 0.01 ❤❤❤❤❤❤❤❤ ❤s P(B′|A2) = 0.99

Fewer false positives would significantly increase the accuracy. Balance that against what is preferable: A false positive or a false negative?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Conditional Probability