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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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Expected values for a game of chance

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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?\(\$\)_____

Calculate Probability of Odd Number: Calculate the probability of spinning an odd number. $\newline$ Since there are $50$ sections and half of them are odd (1, 3, 5, \.\.\., 49), there are $25$ odd numbers. $\newline$ Probability of spinning an odd number ( $P_{\text{odd}}$ ) = Number of odd numbers / Total number of sections $\newline$ $P_{\text{odd}} = \frac{25}{50}$ $\newline$ $P_{\text{odd}} = 0.5$
Calculate Expected Payoff for Odd Number: Calculate the expected payoff for spinning an odd number. $\newline$ The amount won for an odd number is $\$11$ . $\newline$ Expected payoff for an odd number = $P_{\text{odd}} \times \text{Amount won for odd number}$ $\newline$ Expected payoff for an odd number = $0.5 \times \$11$ $\newline$ Expected payoff for an odd number = $\$5.5$
Calculate Probability of Even Number: Calculate the probability of spinning an even number. $\newline$ Since there are $50$ sections and half of them are even ( $2, 4, 6, ..., 50$ ), there are $25$ even numbers. $\newline$ Probability of spinning an even number ( $P_{\text{even}}$ ) = Number of even numbers / Total number of sections $\newline$ $P_{\text{even}} = \frac{25}{50}$ $\newline$ $P_{\text{even}} = 0.5$
Calculate Expected Payoff for Even Number: Calculate the expected payoff for spinning an even number. $\newline$ The amount won for an even number is $\$0$ . $\newline$ Expected payoff for an even number = $P_{\text{even}} \times \text{Amount won for even number}$ $\newline$ Expected payoff for an even number = $0.5 \times \$0$ $\newline$ Expected payoff for an even number = $\$0$
Calculate Total Expected Payoff: Calculate the total expected payoff. $\newline$ Total expected payoff = Expected payoff for odd number + Expected payoff for even number $\newline$ Total expected payoff = $\$5.5$ + $\$0$ $\newline$ Total expected payoff = $\$5.5$

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