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 die is rolled, and a coin is tossed. If the result is a "six" and "heads," you get $\$ 10 .$ For any other result you pay \$1.

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Key Concepts

Fair Game

A game is considered fair if its expected value is zero, meaning that the average gain and loss balance out over time. In other words, neither the player nor the house has a long-term advantage, and the payouts are adjusted such that there is no net profit or loss on average.

Expected Value

The expected value is a fundamental concept in probability that involves calculating the average outcome of a game or random process over many repetitions. It is determined by multiplying each outcome by its probability and summing these products. This concept is critical for assessing whether a game offers a net gain, loss, or break-even in the long run.

Probability

Probability is the measure of how likely an event is to occur and is used to assign weights to different outcomes in a game. Calculating the probability of specific outcomes requires an understanding of the total possible outcomes, which forms the basis for further analysis such as determining expected value.

Independent Events

Independent events are two or more events whose outcomes do not affect each other. When analyzing games that involve multiple components, like separate random experiments, it is important to recognize and utilize the independence of events to accurately compute the combined probabilities.

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