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Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion
MATH 105: Finite Mathematics8-3: 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 newprobability formulas, but rather with an application.
Expected Value
The expected value of a game or procedure is the average value ofa single instance of the procedure if the procedure is repeatedmany 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 newprobability formulas, but rather with an application.
Expected Value
The expected value of a game or procedure is the average value ofa single instance of the procedure if the procedure is repeatedmany 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 newprobability formulas, but rather with an application.
Expected Value
The expected value of a game or procedure is the average value ofa single instance of the procedure if the procedure is repeatedmany 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 newprobability formulas, but rather with an application.
Expected Value
The expected value of a game or procedure is the average value ofa single instance of the procedure if the procedure is repeatedmany 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 newprobability formulas, but rather with an application.
Expected Value
The expected value of a game or procedure is the average value ofa single instance of the procedure if the procedure is repeatedmany 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 playerloses. If a 6 is rolled, the player draws a ball from an urncontaining 2 red and 3 white balls. If it costs $1.00 to play thegame, 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 playerloses. If a 6 is rolled, the player draws a ball from an urncontaining 2 red and 3 white balls. If it costs $1.00 to play thegame, what is the expected value?
Value Probability Product
-1 56 = 25
30 −2530
9 16 ·
35 = 3
302730
99 16 ·
25 = 2
3019830
20030 ≈ $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 draw2 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 draw2 coins at random. What is the expected value of the coins?
Value Probability Product
10 C(3,2)C(9,2) = 3
363036
15 C(3,1)C(4,1)C(9,2) = 12
3618036
20 C(4,2)C(9,2) = 6
3612036
30 C(2,1)C(3,1)C(9,2) = 6
3618036
35 C(2,1)C(4,1)C(9,2) = 8
3628036
50 C(2,2)C(9,2) = 1
365036
84036 ≈ $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 5free-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 5free-throws. What is her expected number of made shots?
Value Probability Product
0 C (5, 0)(.8)0(.2)5 ≈ .00032 0
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 shooterwho 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 successis p, the expected number of success is n · p
Example
If our basketball player took 10 shots, we would expect her tomake .8 · 10 = 8 of them, and if she took 20 shots, we wouldexpect 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 shooterwho 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 successis p, the expected number of success is n · p
Example
If our basketball player took 10 shots, we would expect her tomake .8 · 10 = 8 of them, and if she took 20 shots, we wouldexpect 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 shooterwho 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 successis p, the expected number of success is n · p
Example
If our basketball player took 10 shots, we would expect her tomake .8 · 10 = 8 of them, and if she took 20 shots, we wouldexpect 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 withprobability 0.20. A quality control test examines widgets as theycome off the assembly line. If 50 widgets are checked, how manywould 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 values2 the probability of each value3 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 values2 the probability of each value3 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 values2 the probability of each value3 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 values2 the probability of each value3 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 values2 the probability of each value3 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 lookat statistics. Statistics is the study of ways to represent data withgraphs and numbers and to make inferences about a largepopulation 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 lookat statistics. Statistics is the study of ways to represent data withgraphs and numbers and to make inferences about a largepopulation 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
