25-30= A Fair Game? A game of chance is described. (a) Is...

$25-30=$ A Fair Game? A game of chance is described. (a) Is
the game fair? (b) If the game is not fair, find the payout amount
that would make the game fair.

A die is rolled. You get $\$ 20$ if a one or a six shows; other-
wise, you pay $\$ 10 .$

Key Concepts

Expected Value

Expected value is a key concept in probability and statistics that represents the average outcome of a random event when it is repeated many times. It is calculated by multiplying each outcome by its probability and then summing all these products. In games of chance, the expected value helps determine whether a game is favorable, neutral, or unfavorable to the player over the long run.

Fair Game

A fair game is one in which the expected value for the player is zero, meaning that, over time, the average net gain or loss is zero. In a fair game, the amount won or lost balances out exactly according to the probabilities and payoffs. This concept is central to understanding risk and reward in gambling and economic decision-making.

Probability Distribution

A probability distribution defines the likelihood of each possible outcome of a random process. In the context of a game of chance, it assigns probabilities to each event that can occur, such as the different outcomes of a die roll. Understanding the distribution is essential for determining the expected value and, consequently, whether a game is fair.

Payoff Adjustment

Payoff adjustment involves modifying the amount awarded or charged for specific outcomes to achieve a particular objective, such as making a game fair. By adjusting the payout values, one can alter the expected value of the game to ensure that there is no net gain or loss over time. This concept is often used to design balanced games or to calibrate betting systems.

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