Expected Value If the outcomes of an experiment are given numerical values (such as the

Chapter 9, Problem 9.1.1.194

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Expected Value

If the outcomes of an experiment are given numerical values (such as the total on a roll of two dice, or the payoff on a lottery ticket), we define the expected value to be the sum of all the numerical values times their respective probabilities.

For example, suppose we roll a fair die. If we roll a multiple of 3, we win $3; otherwise we lose $1. The probabilities of the two possible payoffs are shown in the table below:

The expected value is \(3 \times(2 / 6)+(-1) \times(4 / 6)=(6 / 6)-(4 / 6)=1 / 3\).

We interpret this to mean that we would win an average of 1/3 dollar per game in the long run.

(a) A game is called fair if the expected value of the payoff is zero. Assuming that we still win $3 for a multiple of 3, what should we pay for any other outcome in order to make the game fair?

(b) Suppose we roll two fair dice and look at the total under the original rules. That is, we win $3 for rolling a multiple of 3 and lose $1 otherwise. What is the expected value of this game?