A game of chance is described. a. Is the game fair? b. If...

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 card is drawn from a deck. If the card is the ace of spades, you get a payout of $\$ 12 .$ If the card is not an ace, you have to pay $\$ 0.50$.

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Key Concepts

Fair Game

A fair game is one in which the expected value for all participants is zero, meaning that over many iterations, neither party gains or loses money on average. This concept ensures that the probabilities and payouts are structured so that the average outcome balances out.

Expected Value

Expected value is a statistical measure used to determine the long-term average outcome of a random event. It is calculated by multiplying each possible outcome by its probability and then summing these products, providing a single value that represents the average gain or loss expected in the game over many trials.

Probability

Probability is a branch of mathematics that quantifies the likelihood that a specific event will occur. In the context of games, it is used to assign a numerical chance to each outcome, which is essential for calculating the expected value and determining the fairness of the game.

Payoff Adjustment

Payoff adjustment involves modifying the gains or losses associated with different outcomes to achieve a desired balance, such as making a game fair. This concept is critical when the original payouts result in an imbalanced or unfair expected value, requiring a change in rewards or penalties to neutralize any inherent advantage.

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