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At a carnival, there is a game where you can draw one of 5050 balls from a bucket. The balls are numbered from 11 to 5050. If the number on the ball is odd, you win $26\$26. If the number on the ball is even, you win nothing. If you play the game, what is the expected payoff?\newline$\$____

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Q. At a carnival, there is a game where you can draw one of 5050 balls from a bucket. The balls are numbered from 11 to 5050. If the number on the ball is odd, you win $26\$26. If the number on the ball is even, you win nothing. If you play the game, what is the expected payoff?\newline$\$____
  1. Total Balls Count: There are 5050 balls total, half odd and half even. So, there's 2525 odd balls and 2525 even balls.
  2. Odd Ball Prize: You win $26\$26 for an odd ball. The chance of drawing an odd ball is 2525 out of 5050, or 12\frac{1}{2}.
  3. Expected Payoff for Odd Ball: Expected payoff for an odd ball is $26\$26 times the probability of drawing an odd ball, which is 12\frac{1}{2}.\newlineSo, $26×(12)=$13\$26 \times \left(\frac{1}{2}\right) = \$13.
  4. Even Ball Prize: You win $0\$0 for an even ball. The chance of drawing an even ball is also 12\frac{1}{2}.
  5. Expected Payoff for Even Ball: Expected payoff for an even ball is $0\$0 times the probability of drawing an even ball, which is 12\frac{1}{2}.\newlineSo, $0×(12)=$0\$0 \times \left(\frac{1}{2}\right) = \$0.
  6. Total Expected Payoff: Add the expected payoffs for odd and even balls to get the total expected payoff.\newline$13\$13 (for odd) + $0\$0 (for even) = $13\$13.

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