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Probability from Counting Examples (Lots and Lots of Them!) Conclusion MATH 105: Finite Mathematics 7-3: Probability from Counting Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Probability from Counting Examples (Lots and

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

MATH 105: Finite Mathematics 7-3: Probability from Counting

Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Outline

1

Probability from Counting

2

Examples (Lots and Lots of Them!)

3

Conclusion

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Outline

1

Probability from Counting

2

Examples (Lots and Lots of Them!)

3

Conclusion

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Counting and Probability

We have seen the following probability formula used quite often in the last two sections. Probability of Equally Likely Outcomes if E is an event in a sample space S and outcomes in S are all equally likely, then Pr[E] = c(E) c(S) Counting Rules We can use counting rules such as P(n, r) and C(n, r) to find c(E) and c(S).

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Counting and Probability

We have seen the following probability formula used quite often in the last two sections. Probability of Equally Likely Outcomes if E is an event in a sample space S and outcomes in S are all equally likely, then Pr[E] = c(E) c(S) Counting Rules We can use counting rules such as P(n, r) and C(n, r) to find c(E) and c(S).

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Counting and Probability

We have seen the following probability formula used quite often in the last two sections. Probability of Equally Likely Outcomes if E is an event in a sample space S and outcomes in S are all equally likely, then Pr[E] = c(E) c(S) Counting Rules We can use counting rules such as P(n, r) and C(n, r) to find c(E) and c(S).

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Outline

1

Probability from Counting

2

Examples (Lots and Lots of Them!)

3

Conclusion

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Selecting a Subgroup of People

Example A group of 6 women and 5 men wish to select 3 people to perform some task. They decide to draw names out of a hat.

1 What is the probability that all 3 are women? 2 What is the probability of 2 women and 1 man? 3 What is the probability of more women than men? 4 What is the probability of at least one man?

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Selecting a Subgroup of People

Example A group of 6 women and 5 men wish to select 3 people to perform some task. They decide to draw names out of a hat.

1 What is the probability that all 3 are women? 2 What is the probability of 2 women and 1 man? 3 What is the probability of more women than men? 4 What is the probability of at least one man?

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Selecting a Subgroup of People

Example A group of 6 women and 5 men wish to select 3 people to perform some task. They decide to draw names out of a hat.

1 What is the probability that all 3 are women?

C(6, 3) C(11, 3) = 20 165 ≈ 0.121

2 What is the probability of 2 women and 1 man? 3 What is the probability of more women than men? 4 What is the probability of at least one man?

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Selecting a Subgroup of People

Example A group of 6 women and 5 men wish to select 3 people to perform some task. They decide to draw names out of a hat.

1 What is the probability that all 3 are women?

≈ 0.121

2 What is the probability of 2 women and 1 man? 3 What is the probability of more women than men? 4 What is the probability of at least one man?

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Selecting a Subgroup of People

Example A group of 6 women and 5 men wish to select 3 people to perform some task. They decide to draw names out of a hat.

1 What is the probability that all 3 are women?

≈ 0.121

2 What is the probability of 2 women and 1 man?

C(6, 2)C(5, 1) C(11, 3) = 75 165 ≈ 0.455

3 What is the probability of more women than men? 4 What is the probability of at least one man?

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Selecting a Subgroup of People

Example A group of 6 women and 5 men wish to select 3 people to perform some task. They decide to draw names out of a hat.

1 What is the probability that all 3 are women?

≈ 0.121

2 What is the probability of 2 women and 1 man?

≈ 0.455

3 What is the probability of more women than men? 4 What is the probability of at least one man?

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Selecting a Subgroup of People

Example A group of 6 women and 5 men wish to select 3 people to perform some task. They decide to draw names out of a hat.

1 What is the probability that all 3 are women?

≈ 0.121

2 What is the probability of 2 women and 1 man?

≈ 0.455

3 What is the probability of more women than men?

C(6, 3)C(5, 0) + C(6, 2)C(5, 1) C(11, 3) = 95 165 ≈ 0.576

4 What is the probability of at least one man?

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Selecting a Subgroup of People

Example A group of 6 women and 5 men wish to select 3 people to perform some task. They decide to draw names out of a hat.

1 What is the probability that all 3 are women?

≈ 0.121

2 What is the probability of 2 women and 1 man?

≈ 0.455

3 What is the probability of more women than men?

≈ 0.576

4 What is the probability of at least one man?

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Selecting a Subgroup of People

Example A group of 6 women and 5 men wish to select 3 people to perform some task. They decide to draw names out of a hat.

1 What is the probability that all 3 are women?

≈ 0.121

2 What is the probability of 2 women and 1 man?

≈ 0.455

3 What is the probability of more women than men?

≈ 0.576

4 What is the probability of at least one man?

1 − C(6, 3) C(11, 3) = 1 − 20 165 ≈ 0.879

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

License Plates

Example A license plate is composed of 3 letters followed by 3 digits. If a plate is randomly produced, what is the probability that it contains at least one repeated character? Let E be the event that the license has no repeats It is easier to count E than E c(S) = 263 · 103 = 17, 576, 000 c(E) = 26 · 25 · 24 · 10 · 9 · 8 = 11, 232, 000 Pr[E] = 1 − Pr[E] = 1 − 11232000 17576000 ≈ 0.361

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

License Plates

Example A license plate is composed of 3 letters followed by 3 digits. If a plate is randomly produced, what is the probability that it contains at least one repeated character? Let E be the event that the license has no repeats It is easier to count E than E c(S) = 263 · 103 = 17, 576, 000 c(E) = 26 · 25 · 24 · 10 · 9 · 8 = 11, 232, 000 Pr[E] = 1 − Pr[E] = 1 − 11232000 17576000 ≈ 0.361

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

License Plates

Example A license plate is composed of 3 letters followed by 3 digits. If a plate is randomly produced, what is the probability that it contains at least one repeated character? Let E be the event that the license has no repeats It is easier to count E than E c(S) = 263 · 103 = 17, 576, 000 c(E) = 26 · 25 · 24 · 10 · 9 · 8 = 11, 232, 000 Pr[E] = 1 − Pr[E] = 1 − 11232000 17576000 ≈ 0.361

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

License Plates

Example A license plate is composed of 3 letters followed by 3 digits. If a plate is randomly produced, what is the probability that it contains at least one repeated character? Let E be the event that the license has no repeats It is easier to count E than E c(S) = 263 · 103 = 17, 576, 000 c(E) = 26 · 25 · 24 · 10 · 9 · 8 = 11, 232, 000 Pr[E] = 1 − Pr[E] = 1 − 11232000 17576000 ≈ 0.361

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

License Plates

Example A license plate is composed of 3 letters followed by 3 digits. If a plate is randomly produced, what is the probability that it contains at least one repeated character? Let E be the event that the license has no repeats It is easier to count E than E c(S) = 263 · 103 = 17, 576, 000 c(E) = 26 · 25 · 24 · 10 · 9 · 8 = 11, 232, 000 Pr[E] = 1 − Pr[E] = 1 − 11232000 17576000 ≈ 0.361

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

License Plates

Example A license plate is composed of 3 letters followed by 3 digits. If a plate is randomly produced, what is the probability that it contains at least one repeated character? Let E be the event that the license has no repeats It is easier to count E than E c(S) = 263 · 103 = 17, 576, 000 c(E) = 26 · 25 · 24 · 10 · 9 · 8 = 11, 232, 000 Pr[E] = 1 − Pr[E] = 1 − 11232000 17576000 ≈ 0.361

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Casting a Play

Example A play requires 3 male and 2 female rules, including that of “mother”. If there are 5 men and 4 women, including Daisy, auditioning for these parts, and the parts are chosen at random, find each probability.

1 the probability that Daisy gets a part 2 the probability that Daisy get the part of “mother”

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Casting a Play

Example A play requires 3 male and 2 female rules, including that of “mother”. If there are 5 men and 4 women, including Daisy, auditioning for these parts, and the parts are chosen at random, find each probability.

1 the probability that Daisy gets a part 2 the probability that Daisy get the part of “mother”

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Casting a Play

Example A play requires 3 male and 2 female rules, including that of “mother”. If there are 5 men and 4 women, including Daisy, auditioning for these parts, and the parts are chosen at random, find each probability.

1 the probability that Daisy gets a part

C(5, 3)C(3, 1) C(5, 3)C(4, 2) = C(3, 1) C(4, 2) = 3 6 = 0.500

2 the probability that Daisy get the part of “mother”

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Casting a Play

Example A play requires 3 male and 2 female rules, including that of “mother”. If there are 5 men and 4 women, including Daisy, auditioning for these parts, and the parts are chosen at random, find each probability.

1 the probability that Daisy gets a part

C(5, 3)C(3, 1) C(5, 3)C(4, 2) = C(3, 1) C(4, 2) = 3 6 = 0.500

2 the probability that Daisy get the part of “mother”

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Casting a Play

Example A play requires 3 male and 2 female rules, including that of “mother”. If there are 5 men and 4 women, including Daisy, auditioning for these parts, and the parts are chosen at random, find each probability.

1 the probability that Daisy gets a part

C(5, 3)C(3, 1) C(5, 3)C(4, 2) = C(3, 1) C(4, 2) = 3 6 = 0.500

2 the probability that Daisy get the part of “mother”

P(5, 3)P(3, 1) P(5, 3)P(4, 2) = P(3, 1) P(4, 2) = 3 8 = 0.375

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Standing in a Row

Example A family photo of a six-person family is to be taken. If the family members line up randomly in a straight line, what is the probability that the mother and father stand next to each other? Use the “combined-person” concept with 5 people including F-M Don’t forget to count both F-M and M-F

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Standing in a Row

Example A family photo of a six-person family is to be taken. If the family members line up randomly in a straight line, what is the probability that the mother and father stand next to each other? Use the “combined-person” concept with 5 people including F-M Don’t forget to count both F-M and M-F

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Standing in a Row

Example A family photo of a six-person family is to be taken. If the family members line up randomly in a straight line, what is the probability that the mother and father stand next to each other? Use the “combined-person” concept with 5 people including F-M P(5, 5) P(6, 6) = 1 6 ≈ 0.166 Don’t forget to count both F-M and M-F

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Standing in a Row

Example A family photo of a six-person family is to be taken. If the family members line up randomly in a straight line, what is the probability that the mother and father stand next to each other? Use the “combined-person” concept with 5 people including F-M 2 · P(5, 5) P(6, 6) = 2 6 ≈ 0.333 Don’t forget to count both F-M and M-F

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush 2 A full house 3 four of a kind 4 three of a kind 5 two pair 6 a pair

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush 2 A full house 3 four of a kind 4 three of a kind 5 two pair 6 a pair

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush

2 · C(26, 4) C(52, 5) = 29900 2598960 ≈ 0.0115

2 A full house 3 four of a kind 4 three of a kind 5 two pair 6 a pair

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush ≈ 0.0115 2 A full house 3 four of a kind 4 three of a kind 5 two pair 6 a pair

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush ≈ 0.0115 2 A full house

13 · C(4, 3) · 12 · C(4, 2) C(52, 5) = 3744 2598960 ≈ 0.0014

3 four of a kind 4 three of a kind 5 two pair 6 a pair

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush ≈ 0.0115 2 A full house ≈ 0.0014 3 four of a kind 4 three of a kind 5 two pair 6 a pair

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush ≈ 0.0115 2 A full house ≈ 0.0014 3 four of a kind

13 · C(4, 4) · C(48, 1) C(52, 5) = 624 2598960 ≈ 0.00024

4 three of a kind 5 two pair 6 a pair

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush ≈ 0.0115 2 A full house ≈ 0.0014 3 four of a kind ≈ 0.0002 4 three of a kind 5 two pair 6 a pair

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush ≈ 0.0115 2 A full house ≈ 0.0014 3 four of a kind ≈ 0.0002 4 three of a kind

13 · C(4, 3) · C(12, 2) · C(4, 1) · C(4, 1) C(52, 5) = 54912 2598960 ≈ .0211

5 two pair 6 a pair

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush ≈ 0.0115 2 A full house ≈ 0.0014 3 four of a kind ≈ 0.0002 4 three of a kind ≈ 0.0211 5 two pair 6 a pair

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush ≈ 0.0115 2 A full house ≈ 0.0014 3 four of a kind ≈ 0.0002 4 three of a kind ≈ 0.0211 5 two pair

C(13, 2)C(4, 2)C(4, 2)C(48, 1) C(52, 5) = 134784 2598960 ≈ 0.0519

6 a pair

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush ≈ 0.0115 2 A full house ≈ 0.0014 3 four of a kind ≈ 0.0002 4 three of a kind ≈ 0.0211 5 two pair ≈ 0.0519 6 a pair

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Playing Cards

Example Find the probability of each poker hand.

1 A flush ≈ 0.0115 2 A full house ≈ 0.0014 3 four of a kind ≈ 0.0002 4 three of a kind ≈ 0.0211 5 two pair ≈ 0.0519 6 a pair

C(13, 1)C(4, 2)C(12, 3)C(4, 1)C(4, 1)C(4, 1) C(52, 5) ≈ 0.4226

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Flipping a Coin

Example A fair coin is tossed six times.

1 Find the probability exactly two tails appear. 2 Find the probability no more than two tails appear.

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Flipping a Coin

Example A fair coin is tossed six times.

1 Find the probability exactly two tails appear. 2 Find the probability no more than two tails appear.

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Flipping a Coin

Example A fair coin is tossed six times.

1 Find the probability exactly two tails appear.

C(6, 2) 26 = 15 64 ≈ 0.2344

2 Find the probability no more than two tails appear.

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Flipping a Coin

Example A fair coin is tossed six times.

1 Find the probability exactly two tails appear.

≈ 0.2344

2 Find the probability no more than two tails appear.

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Flipping a Coin

Example A fair coin is tossed six times.

1 Find the probability exactly two tails appear.

≈ 0.2344

2 Find the probability no more than two tails appear.

C(6, 0) + C(6, 1) + C(6, 2) 26 = 22 65 ≈ 0.344

SLIDE 50

Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Outline

1

Probability from Counting

2

Examples (Lots and Lots of Them!)

3

Conclusion

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Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Important Concepts

Things to Remember from Section 7-3

1 When dealing with equally likely events, remember:

Pr[E] = c(E) c(S)

2 Use Permutations and Combinations to find c(E) and c(S). 3 Always ask yourself: 1

Does order matter? (Yes: P, No: C)

Am I done producing an event? (Yes: Add, No: Multiply)

SLIDE 52

Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Important Concepts

Things to Remember from Section 7-3

1 When dealing with equally likely events, remember:

Pr[E] = c(E) c(S)

2 Use Permutations and Combinations to find c(E) and c(S). 3 Always ask yourself: 1

Does order matter? (Yes: P, No: C)

Am I done producing an event? (Yes: Add, No: Multiply)

SLIDE 53

Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Important Concepts

Things to Remember from Section 7-3

1 When dealing with equally likely events, remember:

Pr[E] = c(E) c(S)

2 Use Permutations and Combinations to find c(E) and c(S). 3 Always ask yourself: 1

Does order matter? (Yes: P, No: C)

Am I done producing an event? (Yes: Add, No: Multiply)

SLIDE 54

Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Important Concepts

Things to Remember from Section 7-3

1 When dealing with equally likely events, remember:

Pr[E] = c(E) c(S)

2 Use Permutations and Combinations to find c(E) and c(S). 3 Always ask yourself: 1

Does order matter? (Yes: P, No: C)

Am I done producing an event? (Yes: Add, No: Multiply)

SLIDE 55

Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Important Concepts

Things to Remember from Section 7-3

1 When dealing with equally likely events, remember:

Pr[E] = c(E) c(S)

2 Use Permutations and Combinations to find c(E) and c(S). 3 Always ask yourself: 1

Does order matter? (Yes: P, No: C)

Am I done producing an event? (Yes: Add, No: Multiply)

SLIDE 56

Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Important Concepts

Things to Remember from Section 7-3

1 When dealing with equally likely events, remember:

Pr[E] = c(E) c(S)

2 Use Permutations and Combinations to find c(E) and c(S). 3 Always ask yourself: 1

Does order matter? (Yes: P, No: C)

Am I done producing an event? (Yes: Add, No: Multiply)

SLIDE 57

Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Next Time. . .

Now that we have used the tools learned in chapter 6 to compute basic probabilities in chapter 7, it is a good time to review what we’ve covered and assess how much you’ve learned. For next time Review Sections 6-1 through 7-3 (omit 6-6) Prepare for Exam on Friday

SLIDE 58

Probability from Counting Examples (Lots and Lots of Them!) Conclusion

Next Time. . .

Now that we have used the tools learned in chapter 6 to compute basic probabilities in chapter 7, it is a good time to review what we’ve covered and assess how much you’ve learned. For next time Review Sections 6-1 through 7-3 (omit 6-6) Prepare for Exam on Friday