There are $10$ cards in a hat, numbered $1$ to $10$ . The game is to draw one card out of the hat. If the number you draw is odd, you win $27. If the number you draw 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

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Q. There are

10

cards in a hat, numbered

1

10

. The game is to draw one card out of the hat. If the number you draw is odd, you win $27. If the number you draw is even, you win nothing. If you play the game, what is the expected payoff?

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\$

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Odd numbers in the hat: There's $5$ odd numbers ( $1$ , $3$ , $5$ , $7$ , $9$ ) and $5$ even numbers ( $2$ , $4$ , $6$ , $1$ $0$ , $1$ $1$ ) in the hat.
Chance to draw odd number: The chance to draw an odd number is $5$ out of $10$ , or $\frac{1}{2}$ .
Expected payoff for odd number: If you draw an odd number, you win $\$27$ . So, the expected payoff for an odd number is $\frac{1}{2} \times \$27$ .
Total expected payoff: Expected payoff for an odd number is $\$\frac{27}{2}$ , which is $\$13.50$ .
Total expected payoff: Expected payoff for an odd number is $\$\frac{27}{2}$ , which is $\$13.50$ .The chance to draw an even number is also $\frac{5}{10}$ , or $\frac{1}{2}$ .
Total expected payoff: Expected payoff for an odd number is $\$\frac{27}{2}$ , which is $\$13.50$ .The chance to draw an even number is also $\frac{5}{10}$ , or $\frac{1}{2}$ .If you draw an even number, you win nothing, $\$0$ . So, the expected payoff for an even number is $\frac{1}{2} \times \$0$ .
Total expected payoff: Expected payoff for an odd number is $\$\frac{27}{2}$ , which is $\$13.50$ .The chance to draw an even number is also $\frac{5}{10}$ , or $\frac{1}{2}$ .If you draw an even number, you win nothing, $\$0$ . So, the expected payoff for an even number is $\frac{1}{2} \times \$0$ .Expected payoff for an even number is $\$\frac{0}{2}$ , which is $\$0$ .
Total expected payoff: Expected payoff for an odd number is $\$\frac{27}{2}$ , which is $\$13.50$ .The chance to draw an even number is also $\frac{5}{10}$ , or $\frac{1}{2}$ .If you draw an even number, you win nothing, $\$0$ . So, the expected payoff for an even number is $\frac{1}{2} \times \$0$ .Expected payoff for an even number is $\$\frac{0}{2}$ , which is $\$0$ .Add the expected payoffs for odd and even numbers to get the total expected payoff: $\$13.50 + \$0$ .
Total expected payoff: Expected payoff for an odd number is $\$\frac{27}{2}$ , which is $\$13.50$ .The chance to draw an even number is also $5$ out of $10$ , or $\frac{1}{2}$ .If you draw an even number, you win nothing, $\$0$ . So, the expected payoff for an even number is $\frac{1}{2} \times \$0$ .Expected payoff for an even number is $\$\frac{0}{2}$ , which is $\$0$ .Add the expected payoffs for odd and even numbers to get the total expected payoff: $\$13.50 + \$0$ .Total expected payoff is $\$13.50$ .