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At a carnival, there is a game where you can draw one of $50$ balls from a bucket. The balls are numbered from $1$ to $50$ . If the number on the ball is odd, you win $\$26$ . 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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Expected values for a game of chance

Full solution

Q. At a carnival, there is a game where you can draw one of

50

balls from a bucket. The balls are numbered from

1

to

50

. If the number on the ball is odd, you win

\$26

. If the number on the ball is even, you win nothing. If you play the game, what is the expected payoff?

\newline

\$

____

Total Balls Count: There are $50$ balls total, half odd and half even. So, there's $25$ odd balls and $25$ even balls.
Odd Ball Prize: You win $\$26$ for an odd ball. The chance of drawing an odd ball is $25$ out of $50$ , or $\frac{1}{2}$ .
Expected Payoff for Odd Ball: Expected payoff for an odd ball is $\$26$ times the probability of drawing an odd ball, which is $\frac{1}{2}$ . $\newline$ So, $\$26 \times \left(\frac{1}{2}\right) = \$13$ .
Even Ball Prize: You win $\$0$ for an even ball. The chance of drawing an even ball is also $\frac{1}{2}$ .
Expected Payoff for Even Ball: Expected payoff for an even ball is $\$0$ times the probability of drawing an even ball, which is $\frac{1}{2}$ . $\newline$ So, $\$0 \times \left(\frac{1}{2}\right) = \$0$ .
Total Expected Payoff: Add the expected payoffs for odd and even balls to get the total expected payoff. $\newline$ $\$13$ (for odd) + $\$0$ (for even) = $\$13$ .

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