Introduction to Indepencence Examples Conclusion

MATH 105: Finite Mathematics7-5: 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 eventcan effect the probability of another event.

Example

A survey of pizza lovers asked whether the liked thick (C ) or thincrust and extra cheese (E ) or regular. You select a person atrandom. Use the results below to find Pr[C ] and Pr[C |E ].

Extra Cheese No Extra Cheese

Thick Crust 24 16 40Thin Crust 12 8 20

36 24 60

Pr[C ] =40

60=

2

3Pr[C |E ] =

24

36=

2

3

Introduction to Indepencence Examples Conclusion

Conditional Probability

In the last section we saw that knowing something about one eventcan effect the probability of another event.

Example

A survey of pizza lovers asked whether the liked thick (C ) or thincrust and extra cheese (E ) or regular. You select a person atrandom. Use the results below to find Pr[C ] and Pr[C |E ].

Extra Cheese No Extra Cheese

Thick Crust 24 16 40Thin Crust 12 8 20

36 24 60

Pr[C ] =40

60=

2

3Pr[C |E ] =

24

36=

2

3

Introduction to Indepencence Examples Conclusion

Conditional Probability

In the last section we saw that knowing something about one eventcan effect the probability of another event.

Example

A survey of pizza lovers asked whether the liked thick (C ) or thincrust and extra cheese (E ) or regular. You select a person atrandom. Use the results below to find Pr[C ] and Pr[C |E ].

Extra Cheese No Extra Cheese

Thick Crust 24 16 40Thin Crust 12 8 20

36 24 60

Pr[C ] =40

60=

2

3Pr[C |E ] =

24

36=

2

3

Introduction to Indepencence Examples Conclusion

Conditional Probability

In the last section we saw that knowing something about one eventcan effect the probability of another event.

Example

A survey of pizza lovers asked whether the liked thick (C ) or thincrust and extra cheese (E ) or regular. You select a person atrandom. Use the results below to find Pr[C ] and Pr[C |E ].

Extra Cheese No Extra Cheese

Thick Crust 24 16 40Thin Crust 12 8 20

36 24 60

Pr[C ] =40

60=

2

3Pr[C |E ] =

24

36=

2

3

Introduction to Indepencence Examples Conclusion

Independent Events

It is not always the case that information about one event changesthe probability of another event.

Independent Events

Events E and F are called independent if the probability of one isnot changed by having information about the outcome of theother. That is, Pr[E |F ] = Pr[E ]

Tests for Independence

Test for independence using the formula:

Pr[E ∩ F ] = Pr[E ] · Pr[F ]

or, use a Venn Diagram and determine if the ratio of E ∩ F to F isthe 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 changesthe probability of another event.

Independent Events

Events E and F are called independent if the probability of one isnot changed by having information about the outcome of theother. That is, Pr[E |F ] = Pr[E ]

Tests for Independence

Test for independence using the formula:

Pr[E ∩ F ] = Pr[E ] · Pr[F ]

or, use a Venn Diagram and determine if the ratio of E ∩ F to F isthe 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 changesthe probability of another event.

Independent Events

Events E and F are called independent if the probability of one isnot changed by having information about the outcome of theother. That is, Pr[E |F ] = Pr[E ]

Tests for Independence

Test for independence using the formula:

Pr[E ∩ F ] = Pr[E ] · Pr[F ]

or, use a Venn Diagram and determine if the ratio of E ∩ F to F isthe 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] = 416 ,

Pr[B] = 816 , 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] = 416 ,

Pr[B] = 816 , and Pr[A ∩ B] = 2

16 . Are A and B independent?

A B

216

216

616

616

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] = 416 ,

Pr[B] = 816 , and Pr[A ∩ B] = 2

16 . Are A and B independent?

A B

216

216

616

616

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] = 416 ,

Pr[B] = 816 , and Pr[A ∩ B] = 2

16 . Are A and B independent?

A B

216

216

616

616

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] = 416 ,

Pr[B] = 816 , and Pr[A ∩ B] = 2

16 . Are A and B independent?

A B

216

216

616

616

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 tossF : Tails on the second toss

Are E and F independent? Use a tree diagram to find out.

Example

In a group of seeds, 13 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 areindependent.

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 tossF : Tails on the second toss

Are E and F independent? Use a tree diagram to find out.

Example

In a group of seeds, 13 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 areindependent.

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 sedansare blue and the rest are red. A car is picked at random. Are theevents of picking a sedan and picking a blue car independent?

Pr[B] =8

16=

1

2Pr[D] =

6

16=

3

8

Pr[B ∩ D] =5

166= 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 sedansare blue and the rest are red. A car is picked at random. Are theevents of picking a sedan and picking a blue car independent?

Pr[B] =8

16=

1

2Pr[D] =

6

16=

3

8

Pr[B ∩ D] =5

166= 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 sedansare blue and the rest are red. A car is picked at random. Are theevents of picking a sedan and picking a blue car independent?

Pr[B] =8

16=

1

2Pr[D] =

6

16=

3

8

Pr[B ∩ D] =5

166= 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 sedansare blue and the rest are red. A car is picked at random. Are theevents of picking a sedan and picking a blue car independent?

Pr[B] =8

16=

1

2Pr[D] =

6

16=

3

8

Pr[B ∩ D] =5

166= 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 noteasily explored using tree diagrams, such as the final example seenin 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 noteasily explored using tree diagrams, such as the final example seenin section 7-4.

For next time

Read section 8-1