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There is a spinner with `50` equally likely sections, numbered from `1` to `50`. You have the opportunity to spin it. If the number is odd, you win $`11`. If the number is even, you win nothing. If you play the game, what is the expected payoff?\(\$\)_____

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Q. There is a spinner with `50` equally likely sections, numbered from `1` to `50`. You have the opportunity to spin it. If the number is odd, you win $`11`. If the number is even, you win nothing. If you play the game, what is the expected payoff?\(\$\)_____
  1. Calculate Probability of Odd Number: Calculate the probability of spinning an odd number.\newlineSince there are 5050 sections and half of them are odd (1, 3, 5, \.\.\., 49), there are 2525 odd numbers.\newlineProbability of spinning an odd number (PoddP_{\text{odd}}) = Number of odd numbers / Total number of sections\newlinePodd=2550P_{\text{odd}} = \frac{25}{50}\newlinePodd=0.5P_{\text{odd}} = 0.5
  2. Calculate Expected Payoff for Odd Number: Calculate the expected payoff for spinning an odd number.\newlineThe amount won for an odd number is $11\$11.\newlineExpected payoff for an odd number = Podd×Amount won for odd numberP_{\text{odd}} \times \text{Amount won for odd number}\newlineExpected payoff for an odd number = 0.5×$110.5 \times \$11\newlineExpected payoff for an odd number = $5.5\$5.5
  3. Calculate Probability of Even Number: Calculate the probability of spinning an even number.\newlineSince there are 5050 sections and half of them are even (2,4,6,...,502, 4, 6, ..., 50), there are 2525 even numbers.\newlineProbability of spinning an even number (PevenP_{\text{even}}) = Number of even numbers / Total number of sections\newlinePeven=2550P_{\text{even}} = \frac{25}{50}\newlinePeven=0.5P_{\text{even}} = 0.5
  4. Calculate Expected Payoff for Even Number: Calculate the expected payoff for spinning an even number.\newlineThe amount won for an even number is $0\$0.\newlineExpected payoff for an even number = Peven×Amount won for even numberP_{\text{even}} \times \text{Amount won for even number}\newlineExpected payoff for an even number = 0.5×$00.5 \times \$0\newlineExpected payoff for an even number = $0\$0
  5. Calculate Total Expected Payoff: Calculate the total expected payoff.\newlineTotal expected payoff = Expected payoff for odd number + Expected payoff for even number\newlineTotal expected payoff = $5.5\$5.5 + $0\$0\newlineTotal expected payoff = $5.5\$5.5

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