Question

in a carnival game, there is a machine that will shuffle a deck of 100 cards numbered from 1 to 100 and then spit one card out. if the number on the card is even, you win $17. if the number on the card is odd, you win nothing. if you play the game, what is the expected payoff?

Explanation:

Step1: Determine probability of even/odd

Total cards: 100. Even numbers (2,4,...,100): 50. Odd numbers: 50.
Probability of even ($P(\text{even})$) = $\frac{50}{100} = 0.5$.
Probability of odd ($P(\text{odd})$) = $\frac{50}{100} = 0.5$.

Step2: Define payoffs

Payoff for even ($X_{\text{even}}$) = $17$.
Payoff for odd ($X_{\text{odd}}$) = $0$.

Step3: Calculate expected value

Expected payoff $E(X) = P(\text{even}) \cdot X_{\text{even}} + P(\text{odd}) \cdot X_{\text{odd}}$.
Substitute values: $E(X) = 0.5 \cdot 17 + 0.5 \cdot 0$.
Simplify: $E(X) = 8.5 + 0 = 8.5$.

Answer:

$8.5$