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 Introduction to Expected Value 1 Examples 2 Expected Value of a Bernoulli Process 3 Conclusion 4
Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion Outline Introduction to Expected Value 1 Examples 2 Expected Value of a Bernoulli Process 3 Conclusion 4
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 Introduction to Expected Value 1 Examples 2 Expected Value of a Bernoulli Process 3 Conclusion 4
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 6 = 25 5 − 25 -1 30 30 1 6 · 3 5 = 3 27 9 30 30 1 6 · 2 5 = 2 198 99 30 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 C (3 , 2) C (9 , 2) = 3 30 10 36 36 C (3 , 1) C (4 , 1) = 12 180 15 C (9 , 2) 36 36 C (4 , 2) C (9 , 2) = 6 120 20 36 36 C (2 , 1) C (3 , 1) = 6 180 30 36 36 C (9 , 2) C (2 , 1) C (4 , 1) = 8 280 35 C (9 , 2) 36 36 C (2 , 2) C (9 , 2) = 1 50 50 36 36 840 36 ≈ $23 . 33
Introduction to Expected Value Examples Expected Value of a Bernoulli Process Conclusion Outline Introduction to Expected Value 1 Examples 2 Expected Value of a Bernoulli Process 3 Conclusion 4
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 0 0 C (5 , 1)( . 8) 1 ( . 2) 4 ≈ . 0064 1 .0064 C (5 , 2)( . 8) 2 ( . 2) 3 ≈ . 0512 2 .1024 C (5 , 3)( . 8) 3 ( . 2) 2 ≈ . 2048 3 .6144 C (5 , 4)( . 8) 4 ( . 2) 1 ≈ . 4096 4 1.6384 C (5 , 5)( . 8) 5 ( . 2) 0 ≈ . 3277 5 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 Introduction to Expected Value 1 Examples 2 Expected Value of a Bernoulli Process 3 Conclusion 4
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: the possible values 1 the probability of each value 2 the product of value and probability 3
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: the possible values 1 the probability of each value 2 the product of value and probability 3
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: the possible values 1 the probability of each value 2 the product of value and probability 3
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: the possible values 1 the probability of each value 2 the product of value and probability 3
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