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There is a game where the outcome is a random integer from $1$ to $50$ . If the outcome is odd, you win $\$14$ . If the outcome is even, 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 is a game where the outcome is a random integer from

1

to

50

. If the outcome is odd, you win

\$14

. If the outcome is even, you win nothing. If you play the game, what is the expected payoff?

\newline

\$

____

Prompt: question_prompt: What's the expected payoff for playing the game where odd numbers win $\$14$ and even numbers win $\$0$ ?
Total Outcomes: There are $50$ possible outcomes, half odd and half even. So there are $25$ odd numbers and $25$ even numbers from $1$ to $50$ .
Probability Calculation: The probability of getting an odd number is $\frac{25}{50}$ , which simplifies to $\frac{1}{2}$ .
Expected Payoff Calculation: The expected payoff for an odd number is $\$14$ times the probability of getting an odd number. So, $\$14 \times \left(\frac{1}{2}\right)$ .
Final Expected Payoff: Calculating the expected payoff: $\$14 \times \left(\frac{1}{2}\right) = \$7$ .

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