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 card is drawn from a deck, a die is rolled, and a coin is tossed. If the result is the "ace of spades," a "six," and "heads," you get $\$ 600 .$ For any other result you pay $\$ 1 .$

$25-30=$ 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 card is drawn from a deck, a die is rolled, and a coin is
tossed. If the result is the "ace of spades," a "six," and "heads,"
you get $\$ 600 .$ For any other result you pay $\$ 1 .$

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Key Concepts

Compound Probability and Sample Space

When dealing with multiple independent events, the overall probability of a specific combination is calculated by multiplying the probabilities of the individual events. This principle underscores the importance of understanding the sample space, which comprises all possible outcomes, in order to accurately determine the likelihood of a specific result and consequently evaluate the fairness of a game.

Expected Value

Expected value is a statistical measure that calculates the average outcome of a random event if it were repeated many times. It is determined by multiplying each outcome by its probability and summing these products, providing a single value that helps to understand the long-term behavior of the game or decision.

Fair Game

A fair game is one in which the expected value is zero, meaning that the costs and payouts are balanced over the long run, and no player has a built-in advantage. In other words, on average, a player neither gains nor loses money when playing such a game.

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