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There are $10$ cards in a hat, numbered $1$ to $10$ . The game is to draw one card out of the hat. If the number you draw is even, you win $\$20$ . If the number you draw 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 are

10

cards in a hat, numbered

1

to

10

. The game is to draw one card out of the hat. If the number you draw is even, you win

\$20

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

\newline

\$

____

Prompt: question_prompt: What's the expected payoff for drawing an even number from a hat with cards numbered $1$ to $10$ ?
Even Numbers: There are $5$ even numbers ( $2$ , $4$ , $6$ , $8$ , $10$ ) and $5$ odd numbers ( $1$ , $3$ , $5$ , $2$ $0$ , $2$ $1$ ) in the hat.
Probability: The probability of drawing an even number is $5$ out of $10$ , or $\frac{1}{2}$ .
Expected Value - Even: The probability of drawing an odd number is also $5$ out of $10$ , or $\frac{1}{2}$ .
Calculation - Even: If you draw an even number, you win $\$20$ . So the expected value for drawing an even number is $\frac{1}{2} \times \$20$ .
Expected Value - Odd: Calculate the expected value for drawing an even number: $\frac{1}{2} \times (\$20) = (\$10)$ .
Calculation - Odd: If you draw an odd number, you win $\$0$ . So the expected value for drawing an odd number is $\frac{1}{2} \times \$0$ .
Total Expected Payoff: Calculate the expected value for drawing an odd number: $\frac{1}{2} \times (\$0) = (\$0)$ .
Total Expected Payoff: Calculate the expected value for drawing an odd number: $\frac{1}{2} \times (\$0) = (\$0)$ . Add the expected values together to get the total expected payoff: $(\$10) + (\$0) = (\$10)$ .

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