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There are 5050 cards in a hat, numbered 11 to 5050. The game is to draw one card out of the hat. If the number you draw is even, you win $24\$24. If the number you draw is odd, you win nothing. If you play the game, what is the expected payoff?\newline$\$____

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Q. There are 5050 cards in a hat, numbered 11 to 5050. The game is to draw one card out of the hat. If the number you draw is even, you win $24\$24. If the number you draw is odd, you win nothing. If you play the game, what is the expected payoff?\newline$\$____
  1. Identify Even Numbers: There are 2525 even numbers between 11 and 5050, so the chance of drawing an even number is 2525 out of 5050.
  2. Calculate Probability: The probability of drawing an even number is 2550\frac{25}{50}, which simplifies to 12\frac{1}{2}.
  3. Determine Expected Payoff for Even Number: If you draw an even number, you win $24\$24. So, the expected payoff for drawing an even number is 12×$24\frac{1}{2} \times \$24.
  4. Calculate Expected Payoff: Calculating the expected payoff: 12×$(24)=$(12).\frac{1}{2} \times \$(24) = \$(12).
  5. Determine Expected Payoff for Odd Number: Since there's no prize for drawing an odd number, the expected payoff for an odd number is $0\$0.
  6. Calculate Total Expected Payoff: Adding the expected payoffs for even and odd numbers: $12+$0=$12\$12 + \$0 = \$12.

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