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You can play a game with a wheel whose rim is divided into equal sections marked with the numbers from $1$ to $20$ . If you play, the attendant will spin the wheel and a ball will land in a random section. If the number the ball lands on is even, you win $\$18$ . If the number the ball lands on 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

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Q. You can play a game with a wheel whose rim is divided into equal sections marked with the numbers from

1

to

20

. If you play, the attendant will spin the wheel and a ball will land in a random section. If the number the ball lands on is even, you win

\$18

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

\newline

\$

____

Question Prompt: question_prompt: What is the expected payoff for playing the game with the wheel?
Total Wheel Sections: There are $20$ sections on the wheel, so there are $20$ possible outcomes.
Number of Even Numbers: Half of the numbers from $1$ to $20$ are even, which means there are $10$ even numbers.
Probability of Winning: If the ball lands on an even number, you win $\$18$ . So the probability of winning $\$18$ is $10$ out of $20$ , or $\frac{1}{2}$ .
Calculate Expected Payoff: To find the expected payoff, multiply the amount won ( $18) by the probability of winning ($ \frac{ $1$ }{ $2$ }$).
Final Expected Payoff: Expected payoff = \(\$18 \times \frac{1}{2}\)
Final Expected Payoff: Expected payoff = \(\$18 \times \frac{1}{2}\) Expected payoff = \(\$9\)

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