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You are presented with a deck of $20$ cards, numbered from $1$ to $20$ and can draw one card. If the number is odd, you win $\$26$ . If the number 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. You are presented with a deck of

20

cards, numbered from

1

to

20

and can draw one card. If the number is odd, you win

\$26

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

\newline

\$

____

Calculate odd probability: Calculate the probability of drawing an odd number. $\newline$ Number of odd cards: $10$ ( $1$ , $3$ , $5$ , ..., $19$ ) $\newline$ Total number of cards: $20$ $\newline$ Probability of odd number = Number of odd cards / Total number of cards = $10 / 20 = 0.5$
Calculate even probability: Calculate the probability of drawing an even number. $\newline$ Number of even cards: $10$ ( $2$ , $4$ , $6$ , ..., $20$ ) $\newline$ Total number of cards: $20$ $\newline$ Probability of even number = Number of even cards / Total number of cards = $10 / 20 = 0.5$
Calculate odd expected payoff: Calculate the expected payoff for drawing an odd number. $\newline$ Probability of odd number = $0.5$ $\newline$ Amount won for odd number: $(\$)26$ $\newline$ Expected payoff for odd number = Probability of odd number $\times$ Amount won for odd number = $0.5 \times 26 = (\$)13$
Calculate even expected payoff: Calculate the expected payoff for drawing an even number. $\newline$ Probability of even number = $0.5$ $\newline$ Amount won for even number: $\$0$ $\newline$ Expected payoff for even number = Probability of even number $\times$ Amount won for even number = \(0. $5$ \times $0$ = $\$0$ \)
Calculate total expected payoff: Calculate the total expected payoff. $\newline$ Total expected payoff = Expected payoff for odd number + Expected payoff for even number = $\$13$ + $\$0$ = $\$13$

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