MATH 105: Finite Mathematics 8-3: Expected Value Prof. Jonathan - - PowerPoint PPT Presentation

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

MATH 105: Finite Mathematics 8-3: Expected Value

• Prof. Jonathan Duncan

Walla Walla College

Winter Quarter, 2006

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Outline

1

Introduction to Expected Value

2

Examples

3

Expected Value of a Bernoulli Process

4

Conclusion

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Outline

1

Introduction to Expected Value

2

Examples

3

Expected Value of a Bernoulli Process

4

Conclusion

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

What is Expected Value?

Our last section on probability does not introduce any new probability formulas, but rather with an application. Expected Value The expected value of a game or procedure is the average value of a single instance of the procedure if the procedure is repeated many times.

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

What is Expected Value?

Our last section on probability does not introduce any new probability formulas, but rather with an application. Expected Value The expected value of a game or procedure is the average value of a single instance of the procedure if the procedure is repeated many times.

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

What is Expected Value?

Our last section on probability does not introduce any new probability formulas, but rather with an application. Expected Value The expected value of a game or procedure is the average value of a single instance of the procedure if the procedure is repeated many times. We compute expected value using two items of information:

1 Possible values of the procedure 2 Probability of each value

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

What is Expected Value?

Our last section on probability does not introduce any new probability formulas, but rather with an application. Expected Value The expected value of a game or procedure is the average value of a single instance of the procedure if the procedure is repeated many times. We compute expected value using two items of information:

1 Possible values of the procedure 2 Probability of each value

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

What is Expected Value?

Our last section on probability does not introduce any new probability formulas, but rather with an application. Expected Value The expected value of a game or procedure is the average value of a single instance of the procedure if the procedure is repeated many times. We compute expected value using two items of information:

1 Possible values of the procedure 2 Probability of each value

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Outline

1

Introduction to Expected Value

2

Examples

3

Expected Value of a Bernoulli Process

4

Conclusion

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Take me Out to the Ball Game

Example A game starts by rolling a die. If a 1 through 5 is rolled, the player

• loses. If a 6 is rolled, the player draws a ball from an urn

containing 2 red and 3 white balls. If it costs $1.00 to play the game, what is the expected value?

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Take me Out to the Ball Game

Example A game starts by rolling a die. If a 1 through 5 is rolled, the player

• loses. If a 6 is rolled, the player draws a ball from an urn

containing 2 red and 3 white balls. If it costs $1.00 to play the game, what is the expected value? Value Probability Product

• 1

5 6 = 25 30

−25

30

9

1 6 · 3 5 = 3 30 27 30

99

1 6 · 2 5 = 2 30 198 30 200 30 ≈ $6.67

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Pick your Pocket

Example Your pocket contains 3 nickels, 4 dimes, and 2 quarters. You draw 2 coins at random. What is the expected value of the coins?

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Pick your Pocket

Example Your pocket contains 3 nickels, 4 dimes, and 2 quarters. You draw 2 coins at random. What is the expected value of the coins? Value Probability Product 10

C(3,2) C(9,2) = 3 36 30 36

15

C(3,1)C(4,1) C(9,2)

= 12

36 180 36

20

C(4,2) C(9,2) = 6 36 120 36

30

C(2,1)C(3,1) C(9,2)

= 6

36 180 36

35

C(2,1)C(4,1) C(9,2)

= 8

36 280 36

50

C(2,2) C(9,2) = 1 36 50 36 840 36 ≈ $23.33

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Outline

1

Introduction to Expected Value

2

Examples

3

Expected Value of a Bernoulli Process

4

Conclusion

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Basketball

Example During a basketball game, an 80% free-throw shooter attempts 5 free-throws. What is her expected number of made shots?

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Basketball

Example During a basketball game, an 80% free-throw shooter attempts 5 free-throws. What is her expected number of made shots? Value Probability Product C(5, 0)(.8)0(.2)5 ≈ .00032 1 C(5, 1)(.8)1(.2)4 ≈ .0064 .0064 2 C(5, 2)(.8)2(.2)3 ≈ .0512 .1024 3 C(5, 3)(.8)3(.2)2 ≈ .2048 .6144 4 C(5, 4)(.8)4(.2)1 ≈ .4096 1.6384 5 C(5, 5)(.8)5(.2)0 ≈ .3277 1.6384 4

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Basketball Part II

Note: In the previous example, we saw that an 80% free-throw shooter who takes 5 shots expects to make 4 of them. Is this a surprise? Expected Value of a Bernoulli Process In a Bernoulli Process of n trials where the probability of a success is p, the expected number of success is n · p Example If our basketball player took 10 shots, we would expect her to make .8 · 10 = 8 of them, and if she took 20 shots, we would expect her to make .8 · 20 = 16, and so on.

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Basketball Part II

Note: In the previous example, we saw that an 80% free-throw shooter who takes 5 shots expects to make 4 of them. Is this a surprise? Expected Value of a Bernoulli Process In a Bernoulli Process of n trials where the probability of a success is p, the expected number of success is n · p Example If our basketball player took 10 shots, we would expect her to make .8 · 10 = 8 of them, and if she took 20 shots, we would expect her to make .8 · 20 = 16, and so on.

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Basketball Part II

Note: In the previous example, we saw that an 80% free-throw shooter who takes 5 shots expects to make 4 of them. Is this a surprise? Expected Value of a Bernoulli Process In a Bernoulli Process of n trials where the probability of a success is p, the expected number of success is n · p Example If our basketball player took 10 shots, we would expect her to make .8 · 10 = 8 of them, and if she took 20 shots, we would expect her to make .8 · 20 = 16, and so on.

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Defective Widgets

Example Defective widgets are produced by a widget factory randomly with probability 0.20. A quality control test examines widgets as they come off the assembly line. If 50 widgets are checked, how many would you expect to be defective? 50(0.20) = 10 defective widgets

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Outline

1

Introduction to Expected Value

2

Examples

3

Expected Value of a Bernoulli Process

4

Conclusion

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Important Concepts

Things to Remember from Section 8-3

1 To find expected value us a table which contains columns for: 1

the possible values

2

the probability of each value

3

the product of value and probability

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Important Concepts

Things to Remember from Section 8-3

1 To find expected value us a table which contains columns for: 1

the possible values

2

the probability of each value

3

the product of value and probability

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Important Concepts

Things to Remember from Section 8-3

1 To find expected value us a table which contains columns for: 1

the possible values

2

the probability of each value

3

the product of value and probability

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Important Concepts

Things to Remember from Section 8-3

1 To find expected value us a table which contains columns for: 1

the possible values

2

the probability of each value

3

the product of value and probability

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Important Concepts

Things to Remember from Section 8-3

1 To find expected value us a table which contains columns for: 1

the possible values

2

the probability of each value

3

the product of value and probability

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Next Time. . .

We are now done with probability. In the next chapter we will look at statistics. Statistics is the study of ways to represent data with graphs and numbers and to make inferences about a large population from a small sample of the population. For next time Read sections 9-1 and 9-2 Prepare for quiz on 8-2 and 8-3

Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion

Next Time. . .

We are now done with probability. In the next chapter we will look at statistics. Statistics is the study of ways to represent data with graphs and numbers and to make inferences about a large population from a small sample of the population. For next time Read sections 9-1 and 9-2 Prepare for quiz on 8-2 and 8-3