Copyright © Cengage Learning. All rights reserved. Probability and Statistics

Copyright © Cengage Learning. All rights reserved Expected Value

3 Objectives ► Expected Value ► What Is a Fair Game?

4 Expected Value

5 Suppose that a coin has probability 0.8 of showing heads. If the coin is tossed many times, we would expect to get heads about 80% of the time. Now, suppose that you get a payout of one dollar for each head. If you play this game many times, you would expect on average to gain $0.80 per game: = $1.00  0.80 = $0.80

6 Expected Value The reasoning in the example discussed motivates the following definition. Mathematical expectation, sometimes stated as the expected value or long-term average, is defined to be the probability of the event multiplied by the value assigned to the event.

7 Example X12345 P(x) Find the expected value of x in the probability distribution below:

8 Example 2 – Finding Expected Value In Monte Carlo the game of roulette is played on a wheel with slots numbered 0, 1, 2,..., 36. The wheel is spun, and a ball dropped in the wheel is equally likely to end up in any one of the slots. To play the game, you bet $1 on any number. (For example, you may bet $1 on number 23.) If the ball stops in your slot, you get $36 (the $1 you bet plus $35). Find the expected value of this game.

9 Example 2 – Solution The gambler can gain $35 with probability and can lose $1 with probability. So the gambler’s expected value is In other words, if you play this game many times, you would expect to lose 2.7 cents on every dollar you bet (on average). Consequently, the house expects to gain 2.7 cents on every dollar that is bet.

10 Examples A local church decides to raise money by raffling off a micro oven worth $400, a dinner for two worth $80, and 2 books worth $20. A total of 2000 tickets are sold at $1 each. Find the expected value of winnings for a person who buys 1 ticket in the raffle.

11 Examples Bob is 50 years old. He must decide 1 of 2 options from an insurance company: get $60000 at age 60 or $65000 at age 70. Given that probability of person living from age 50 to 60 is.88, and probability of person living from age 50 to 70 is.64. Use expected value to help Bob to select which option is more valuable.

12 What Is a Fair Game?

13 What Is a Fair Game? A fair game is game with expected value zero. So if you play a fair game many times, you would expect, on average, to break even.

14 Example 4 – A Fair Game? Suppose that you play the following game. A card is drawn from a deck. If the card is an ace, you get a payout of $10. If the card is not an ace, you have to pay $1. (a) Is this a fair game? (b) If the game is not fair, find the payout amount that would make this game a fair game.

15 Example 4(a) – Solution In this game you get a payout of $10 if an ace is drawn (probability ), and you lose $1 if any other card is drawn (probability ). So the expected value is Since the expected value is not zero, the game is not fair. If you play this game many times, you would expect to lose, on average,  $0.15, per game.

16 Example 4(b) – Solution We want to find the payout x that makes the expected value 0. = 0 Solving this equation we get x = 12. So a payout of $12 for an ace would make this a fair game. cont’d

17 What Is a Fair Game? Games of chance in casinos are never fair; the gambler always has a negative expected value This makes gambling profitable for the casino and unprofitable for the gambler.