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There are $50$ cards in a hat, numbered $1$ to $50$ . The game is to draw one card out of the hat. If the number you draw is even, you win $\$24$ . 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

50

cards in a hat, numbered

1

to

50

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

\$24

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

\newline

\$

____

Identify Even Numbers: There are $25$ even numbers between $1$ and $50$ , so the chance of drawing an even number is $25$ out of $50$ .
Calculate Probability: The probability of drawing an even number is $\frac{25}{50}$ , which simplifies to $\frac{1}{2}$ .
Determine Expected Payoff for Even Number: If you draw an even number, you win $\$24$ . So, the expected payoff for drawing an even number is $\frac{1}{2} \times \$24$ .
Calculate Expected Payoff: Calculating the expected payoff: $\frac{1}{2} \times \$(24) = \$(12).$
Determine Expected Payoff for Odd Number: Since there's no prize for drawing an odd number, the expected payoff for an odd number is $\$0$ .
Calculate Total Expected Payoff: Adding the expected payoffs for even and odd numbers: $\$12 + \$0 = \$12$ .

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