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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 $\$12$ . 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

\$12

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

\newline

\$

____

Game Payoff: question_prompt: What's the expected payoff for the game with the cards?
Card Distribution: There's $10$ cards, $5$ are even ( $2$ , $4$ , $6$ , $8$ , $10$ ) and $5$ are odd ( $1$ , $3$ , $5$ , $5$ $1$ , $5$ $2$ ).
Winning Probability: You win $\$12$ for even numbers, so the chance to win is $5$ out of $10$ , or $\frac{1}{2}$ .
Expected Payoff Calculation: Expected payoff is calculated by multiplying the probability of winning $p$ with the amount won $x$ for each outcome, resulting in the formula $p \times x$ .
Expected Payoff: So, the expected payoff is $\frac{1}{2}$ * $\$12$ for the even numbers.
Math Calculation: Doing the math: $\frac{1}{2}$ * $\$12$ = $\$6$ .

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