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.)

Answer: The game is not fair; the expected value is \(-\$0.50\).

Explanation: To determine if the game is fair, we need to calculate the expected value. The expected value is calculated by considering all possible outcomes, their probabilities, and their associated payoffs.

Steps:

  1. Identify the outcomes and probabilities:
  • There are 26 red cards in a deck of 52 cards. The probability of drawing a red card is \(\frac{26}{52} = \frac{1}{2}\).
  • There are 26 black cards in a deck of 52 cards. The probability of drawing a black card is also \(\frac{26}{52} = \frac{1}{2}\).
  1. Calculate the payoffs:
  • If you draw a red card, you double your money. Since you paid $1 to play, you win $2, but considering the cost, your net gain is $1.
  • If you draw a black card, you win nothing, resulting in a net loss of $1 (the cost of playing).
  1. Calculate the expected value:

\[ \text{Expected Value} = \left(\frac{1}{2} \times 1\right) + \left(\frac{1}{2} \times (-1)\right) = \frac{1}{2} - \frac{1}{2} = 0 \]

However, we must consider the cost of playing, which is $1. Therefore, the net expected value is:

\[ \text{Net Expected Value} = 0 - 1 = -\$0.50 \]

Since the expected value is negative, the game is not in your favor, and it is not fair.

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