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At a carnival, there is a game where you can draw one of $20$ balls from a bucket. The balls are numbered from $1$ to $20$ . If the number on the ball is even, you win $\$11$ . If the number on the ball 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. At a carnival, there is a game where you can draw one of

20

balls from a bucket. The balls are numbered from

1

to

20

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

\$11

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

\newline

\$

____

Calculate probability of even: question_prompt: What's the expected payoff for the carnival game where you draw numbered balls from a bucket?
Calculate probability of odd: Step $1$ : Calculate the probability of drawing an even numbered ball. There are $10$ even numbers in $1$ to $20$ , so the probability is $\frac{10}{20}$ or $\frac{1}{2}$ .
Calculate expected payoff for even: Step $2$ : Calculate the probability of drawing an odd numbered ball. There are also $10$ odd numbers in $1$ to $20$ , so the probability is $\frac{10}{20}$ or $\frac{1}{2}$ .
Calculate expected payoff for odd: Step $3$ : Calculate the expected payoff for drawing an even numbered ball. Since the payoff is $\$11$ for an even number, multiply the probability by the payoff: $(1/2) \times \$11 = \$5.50$ .
Find total expected payoff: Step $4$ : Calculate the expected payoff for drawing an odd numbered ball. Since there's no payoff for an odd number, multiply the probability by $0$ : $\frac{1}{2}$ * $\$0$ = $\$0$ .
Find total expected payoff: Step $4$ : Calculate the expected payoff for drawing an odd numbered ball. Since there's no payoff for an odd number, multiply the probability by $0$ : $(\frac{1}{2}) \times (\$0) = (\$0)$ . Step $5$ : Add the expected payoffs together to find the total expected payoff. $(\$5.50)$ (even) + $(\$0)$ (odd) = $(\$5.50)$ .

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