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There is a spinner with $20$ equally likely sections, numbered from $1$ to $20$ . You have the opportunity to spin it. If the number is even, you win $\$17$ . If the number is odd, you win nothing. If you play the game, what is the expected payoff? $\newline$ $\$$ ____

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Expected values for a game of chance

Full solution

Q. There is a spinner with

20

equally likely sections, numbered from

1

to

20

. You have the opportunity to spin it. If the number is even, you win

\$17

. If the number is odd, you win nothing. If you play the game, what is the expected payoff?

\newline

\$

____

Calculate probability of even: Calculate the probability of spinning an even number. There are $10$ even numbers ( $2$ , $4$ , $6$ , $8$ , $10$ , $12$ , $14$ , $16$ , $18$ , $2$ $0$ ) out of $2$ $0$ total numbers. So, $2$ $2$ .
Calculate probability of odd: Calculate the probability of spinning an odd number. There are $10$ odd numbers ( $1$ , $3$ , $5$ , $7$ , $9$ , $11$ , $13$ , $15$ , $17$ , $1$ $0$ ) out of $1$ $1$ total numbers. So, $1$ $2$ .
Calculate expected payoff for even: Calculate the expected payoff for spinning an even number. Expected payoff for even = $P(\text{even}) \times \text{Amount won for even}$ . Expected payoff for even = $\frac{1}{2} \times \$(17)$ .
Calculate expected payoff for odd: Calculate the expected payoff for spinning an odd number. Expected payoff for odd = $P(\text{odd}) \times \text{Amount won for odd}$ . Expected payoff for odd = $\frac{1}{2} \times \$(0)$ .
Add expected payoffs: Add the expected payoffs for even and odd numbers to get the total expected payoff. Total expected payoff = Expected payoff for even + Expected payoff for odd. Total expected payoff = $(\frac{1}{2} * (\$)17) + (\frac{1}{2} * (\$)0)$ .
Perform calculations: Perform the calculations to find the total expected payoff. Total expected payoff = $0.5 * 17$ + $0.5 * 0$ . Total expected payoff = $\$8.5$ + $\$0$ . Total expected payoff = $\$8.5$ .

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