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There are 1010 cards in a hat, numbered 11 to 1010. The game is to draw one card out of the hat. If the number you draw is even, you win $12\$12. If the number you draw is odd, you win nothing. If you play the game, what is the expected payoff?\newline$\$____

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Q. There are 1010 cards in a hat, numbered 11 to 1010. The game is to draw one card out of the hat. If the number you draw is even, you win $12\$12. If the number you draw is odd, you win nothing. If you play the game, what is the expected payoff?\newline$\$____
  1. Game Payoff: question_prompt: What's the expected payoff for the game with the cards?
  2. Card Distribution: There's 1010 cards, 55 are even (22, 44, 66, 88, 1010) and 55 are odd (11, 33, 55, 5511, 5522).
  3. Winning Probability: You win $12\$12 for even numbers, so the chance to win is 55 out of 1010, or 12\frac{1}{2}.
  4. Expected Payoff Calculation: Expected payoff is calculated by multiplying the probability of winning pp with the amount won xx for each outcome, resulting in the formula p×xp \times x.
  5. Expected Payoff: So, the expected payoff is 12\frac{1}{2} * $12\$12 for the even numbers.
  6. Math Calculation: Doing the math: 12\frac{1}{2} * $12\$12 = $6\$6.

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