At a carnival, there is a game where you can draw one of $100$ balls from a bucket. The balls are numbered from $1$ to $100$ . If the number on the ball is even, you win $\$ 13$ . If the number on the ball is odd, you win nothing. If you play the game, what is the expected payoff?

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Expected values for a game of chance

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Q. At a carnival, there is a game where you can draw one of

100

balls from a bucket. The balls are numbered from

1

100

. If the number on the ball is even, you win

\$ 13

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

Calculate probability of even number: Determine the probability of drawing an even-numbered ball. Since there are $100$ balls numbered from $1$ to $100$ , there are $50$ even numbers and $50$ odd numbers. Probability of drawing an even number $= \frac{\text{Number of even numbers}}{\text{Total number of balls}}$ Probability of drawing an even number $= \frac{50}{100}$
Calculate expected payoff for even number: Calculate the expected payoff for drawing an even-numbered ball. $\newline$ The expected payoff for an event is calculated by multiplying the probability of the event by the payoff for that event. $\newline$ Expected payoff for even number = Probability of drawing an even number $\times$ Payoff for even number $\newline$ Expected payoff for even number = $(50 / 100) \times \$13$
Perform expected payoff calculation: Perform the calculation for the expected payoff for drawing an even-numbered ball. $\newline$ Expected payoff for even number = $$0$.$5$ \times \$($13$)$\newline$Expected payoff for even number = \$($6$.$50$)
Calculate probability of odd number: Determine the probability of drawing an odd-numbered ball.$\newline$Since there are $50$ odd numbers out of $100$ balls, the probability is the same as for even numbers.$\newline$Probability of drawing an odd number $= \frac{\text{Number of odd numbers}}{\text{Total number of balls}}$$\newline$Probability of drawing an odd number $= \frac{50}{100}$
Calculate expected payoff for odd number: Calculate the expected payoff for drawing an odd-numbered ball.$\newline$Since the payoff for drawing an odd-numbered ball is $\$0$, the expected payoff is:$\newline$Expected payoff for odd number = Probability of drawing an odd number $*$ Payoff for odd number$\newline$Expected payoff for odd number = $(\(50$ / $100$) $*$ \$$0$\)$\newline$Expected payoff for odd number = $\(0$.$5$ $*$ \$$0$\)$\newline$Expected payoff for odd number = $\$0$
Calculate total expected payoff: Calculate the total expected payoff for playing the game.$\newline$The total expected payoff is the sum of the expected payoffs for each possible outcome.$\newline$Total expected payoff = Expected payoff for even number + Expected payoff for odd number$\newline$Total expected payoff = $\$6.50$ + $\$0$$\newline$Total expected payoff = $\$6.50$