MATH 105: Finite Mathematics 7-5: Independent Events Prof. Jonathan - - PowerPoint PPT Presentation

Introduction to Indepencence Examples Conclusion

MATH 105: Finite Mathematics 7-5: Independent Events

• Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006

Introduction to Indepencence Examples Conclusion

Outline

1

Introduction to Indepencence

2

Examples

3

Conclusion

Introduction to Indepencence Examples Conclusion

Outline

1

Introduction to Indepencence

2

Examples

3

Conclusion

Introduction to Indepencence Examples Conclusion

Conditional Probability

In the last section we saw that knowing something about one event can effect the probability of another event. Example A survey of pizza lovers asked whether the liked thick (C) or thin crust and extra cheese (E) or regular. You select a person at

• random. Use the results below to find Pr[C] and Pr[C|E].

Extra Cheese No Extra Cheese Thick Crust 24 16 40 Thin Crust 12 8 20 36 24 60 Pr[C] = 40 60 = 2 3 Pr[C|E] = 24 36 = 2 3

Introduction to Indepencence Examples Conclusion

Conditional Probability

In the last section we saw that knowing something about one event can effect the probability of another event. Example A survey of pizza lovers asked whether the liked thick (C) or thin crust and extra cheese (E) or regular. You select a person at

• random. Use the results below to find Pr[C] and Pr[C|E].

Extra Cheese No Extra Cheese Thick Crust 24 16 40 Thin Crust 12 8 20 36 24 60 Pr[C] = 40 60 = 2 3 Pr[C|E] = 24 36 = 2 3

Introduction to Indepencence Examples Conclusion

Conditional Probability

In the last section we saw that knowing something about one event can effect the probability of another event. Example A survey of pizza lovers asked whether the liked thick (C) or thin crust and extra cheese (E) or regular. You select a person at

• random. Use the results below to find Pr[C] and Pr[C|E].

Extra Cheese No Extra Cheese Thick Crust 24 16 40 Thin Crust 12 8 20 36 24 60 Pr[C] = 40 60 = 2 3 Pr[C|E] = 24 36 = 2 3

Introduction to Indepencence Examples Conclusion

Conditional Probability

In the last section we saw that knowing something about one event can effect the probability of another event. Example A survey of pizza lovers asked whether the liked thick (C) or thin crust and extra cheese (E) or regular. You select a person at

• random. Use the results below to find Pr[C] and Pr[C|E].

Extra Cheese No Extra Cheese Thick Crust 24 16 40 Thin Crust 12 8 20 36 24 60 Pr[C] = 40 60 = 2 3 Pr[C|E] = 24 36 = 2 3

Introduction to Indepencence Examples Conclusion

Independent Events

It is not always the case that information about one event changes the probability of another event. Independent Events Events E and F are called independent if the probability of one is not changed by having information about the outcome of the

• ther. That is, Pr[E|F] = Pr[E]

Tests for Independence Test for independence using the formula: Pr[E ∩ F] = Pr[E] · Pr[F]

• r, use a Venn Diagram and determine if the ratio of E ∩ F to F is

the same as the ratio of E to S

Introduction to Indepencence Examples Conclusion

Independent Events

It is not always the case that information about one event changes the probability of another event. Independent Events Events E and F are called independent if the probability of one is not changed by having information about the outcome of the

• ther. That is, Pr[E|F] = Pr[E]

Tests for Independence Test for independence using the formula: Pr[E ∩ F] = Pr[E] · Pr[F]

• r, use a Venn Diagram and determine if the ratio of E ∩ F to F is

the same as the ratio of E to S

Introduction to Indepencence Examples Conclusion

Independent Events

It is not always the case that information about one event changes the probability of another event. Independent Events Events E and F are called independent if the probability of one is not changed by having information about the outcome of the

• ther. That is, Pr[E|F] = Pr[E]

Tests for Independence Test for independence using the formula: Pr[E ∩ F] = Pr[E] · Pr[F]

• r, use a Venn Diagram and determine if the ratio of E ∩ F to F is

the same as the ratio of E to S

Introduction to Indepencence Examples Conclusion

Outline

1

Introduction to Indepencence

2

Examples

3

Conclusion

Introduction to Indepencence Examples Conclusion

A Venn Diagram Example

Example Let A and B be events in a sample space S such that Pr[A] = 4

16,

Pr[B] = 8

16, and Pr[A ∩ B] = 2

• 16. Are A and B independent?

Pr[A ∩ B] = 2 16 = 1 8 Pr[A]·Pr[B] = 4 16· 8 16 = 1 8 Independent!

Introduction to Indepencence Examples Conclusion

A Venn Diagram Example

Example Let A and B be events in a sample space S such that Pr[A] = 4

16,

Pr[B] = 8

16, and Pr[A ∩ B] = 2

• 16. Are A and B independent?

A B

2 16 2 16 6 16 6 16

Pr[A ∩ B] = 2 16 = 1 8 Pr[A]·Pr[B] = 4 16· 8 16 = 1 8 Independent!

Introduction to Indepencence Examples Conclusion

A Venn Diagram Example

Example Let A and B be events in a sample space S such that Pr[A] = 4

16,

Pr[B] = 8

16, and Pr[A ∩ B] = 2

• 16. Are A and B independent?

A B

2 16 2 16 6 16 6 16

Pr[A ∩ B] = 2 16 = 1 8 Pr[A]·Pr[B] = 4 16· 8 16 = 1 8 Independent!

Introduction to Indepencence Examples Conclusion

A Venn Diagram Example

Example Let A and B be events in a sample space S such that Pr[A] = 4

16,

Pr[B] = 8

16, and Pr[A ∩ B] = 2

• 16. Are A and B independent?

A B

2 16 2 16 6 16 6 16

Pr[A ∩ B] = 2 16 = 1 8 Pr[A]·Pr[B] = 4 16· 8 16 = 1 8 Independent!

Introduction to Indepencence Examples Conclusion

A Venn Diagram Example

Example Let A and B be events in a sample space S such that Pr[A] = 4

16,

Pr[B] = 8

16, and Pr[A ∩ B] = 2

• 16. Are A and B independent?

A B

2 16 2 16 6 16 6 16

Pr[A ∩ B] = 2 16 = 1 8 Pr[A]·Pr[B] = 4 16· 8 16 = 1 8 Independent!

Introduction to Indepencence Examples Conclusion

Independence and Tree Diagrams

Example A fair coin is tossed twocie and events E and F are defined as: E: Heads on the first toss F: Tails on the second toss Are E and F independent? Use a tree diagram to find out. Example In a group of seeds, 1

3 of which should produce violets, the best

germinateion that can be obtained is 60%. If one seed is planted, what is the probability it will grow a violet? Assume the events are independent. Solve both using a tree diagram and a Venn diagram.

Introduction to Indepencence Examples Conclusion

Independence and Tree Diagrams

Example A fair coin is tossed twocie and events E and F are defined as: E: Heads on the first toss F: Tails on the second toss Are E and F independent? Use a tree diagram to find out. Example In a group of seeds, 1

3 of which should produce violets, the best

germinateion that can be obtained is 60%. If one seed is planted, what is the probability it will grow a violet? Assume the events are independent. Solve both using a tree diagram and a Venn diagram.

Introduction to Indepencence Examples Conclusion

A Used Car Lot

Example There are 16 cars on a used car lot: 10 compacts and 6 sedans. Three of the compacts are blue, the rest are red. Five of the sedans are blue and the rest are red. A car is picked at random. Are the events of picking a sedan and picking a blue car independent? Pr[B] = 8 16 = 1 2 Pr[D] = 6 16 = 3 8 Pr[B ∩ D] = 5 16 = 1 2 · 3 8 These events are not independent.

Introduction to Indepencence Examples Conclusion

A Used Car Lot

Example There are 16 cars on a used car lot: 10 compacts and 6 sedans. Three of the compacts are blue, the rest are red. Five of the sedans are blue and the rest are red. A car is picked at random. Are the events of picking a sedan and picking a blue car independent? Pr[B] = 8 16 = 1 2 Pr[D] = 6 16 = 3 8 Pr[B ∩ D] = 5 16 = 1 2 · 3 8 These events are not independent.

Introduction to Indepencence Examples Conclusion

A Used Car Lot

Example There are 16 cars on a used car lot: 10 compacts and 6 sedans. Three of the compacts are blue, the rest are red. Five of the sedans are blue and the rest are red. A car is picked at random. Are the events of picking a sedan and picking a blue car independent? Pr[B] = 8 16 = 1 2 Pr[D] = 6 16 = 3 8 Pr[B ∩ D] = 5 16 = 1 2 · 3 8 These events are not independent.

Introduction to Indepencence Examples Conclusion

A Used Car Lot

Example There are 16 cars on a used car lot: 10 compacts and 6 sedans. Three of the compacts are blue, the rest are red. Five of the sedans are blue and the rest are red. A car is picked at random. Are the events of picking a sedan and picking a blue car independent? Pr[B] = 8 16 = 1 2 Pr[D] = 6 16 = 3 8 Pr[B ∩ D] = 5 16 = 1 2 · 3 8 These events are not independent.

Introduction to Indepencence Examples Conclusion

Outline

1

Introduction to Indepencence

2

Examples

3

Conclusion

Introduction to Indepencence Examples Conclusion

Important Concepts

Things to Remember from Section 7-5

1 Events A and B are independent if

Pr[A ∩ B] = Pr[A] · Pr[B]

2 Events A and B are independent if

Pr[A|B] = Pr[A]

Introduction to Indepencence Examples Conclusion

Important Concepts

Things to Remember from Section 7-5

1 Events A and B are independent if

Pr[A ∩ B] = Pr[A] · Pr[B]

2 Events A and B are independent if

Pr[A|B] = Pr[A]

Introduction to Indepencence Examples Conclusion

Important Concepts

Things to Remember from Section 7-5

1 Events A and B are independent if

Pr[A ∩ B] = Pr[A] · Pr[B]

2 Events A and B are independent if

Pr[A|B] = Pr[A]

Introduction to Indepencence Examples Conclusion

Next Time. . .

Next time we will explore conditional probabilities which are not easily explored using tree diagrams, such as the final example seen in section 7-4. For next time Read section 8-1

Introduction to Indepencence Examples Conclusion

Next Time. . .

Next time we will explore conditional probabilities which are not easily explored using tree diagrams, such as the final example seen in section 7-4. For next time Read section 8-1