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"A game is said to be fair if the expected value (after considering the cost) is 0. This means that in the long run, both the player and the ""house"" would expect to win nothing. If the value is positive, the game is in your favor. If the value is negative, the game is not in your favor.
At a carnival, you pay $1 to choose a card from a standard deck. If you choose a red card, you double your money, but if you pick a black card, you do not get any.
(A standard deck of cards has 52 cards. 26 of the cards are red.)"
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Answer
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Step 1: Determine the possible outcomes and their probabilities.
P(Black) = frac{26}{52} = frac{1}{2}
There are two possible outcomes in this game: choosing a red card or choosing a black card. A standard deck has 52 cards, with 26 red cards and 26 black cards. Therefore, the probability of choosing a red card is and the probability of choosing a black card is
Step 2: Calculate the expected value of the game.
Expected\ Value = $2 \times frac{1}{2} + $0 \times frac{1}{2} = $1
The expected value is calculated by multiplying the value of each outcome by its probability and summing the results: Substituting these values into the expected value formula, we get:
Final Answer
The expected value of this game is $1. This means that, in the long run, you would expect to win $1 for every game you play. Since the expected value is positive, the game is in your favor. However, it's important to note that this doesn't guarantee a win every time you play; it simply describes the average outcome over many games.
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