There is a spinner with $10$ equally likely sections, numbered from $1$ to $10$ . You have the opportunity to spin it. If the number is odd, you win $\$13$ . 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

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Q. There is a spinner with

10

equally likely sections, numbered from

1

10

. You have the opportunity to spin it. If the number is odd, you win

\$13

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

\newline

\$

____

Probability of landing: There are $10$ sections, so the probability of landing on any section is $\frac{1}{10}$ .
Odd vs Even Sections: $5$ of these sections are odd ( $1, 3, 5, 7, 9$ ), and $5$ are even ( $2, 4, 6, 8, 10$ ).
Expected value for odd number: The probability of landing on an odd number is $\frac{5}{10}$ or $\frac{1}{2}$ .
Expected value for even number: If you land on an odd number, you win $\$13$ , so the expected value for an odd number is $13 \times (1/2)$ .
Total expected payoff: The expected value for an odd number is $\$6.50$ .
Total expected payoff: The expected value for an odd number is $\$6.50$ .The probability of landing on an even number is also $\frac{1}{2}$ , but you win $\$0$ for an even number.
Total expected payoff: The expected value for an odd number is $\$6.50$ .The probability of landing on an even number is also $\frac{1}{2}$ , but you win $\$0$ for an even number.The expected value for an even number is $0 \times \left(\frac{1}{2}\right)$ .
Total expected payoff: The expected value for an odd number is $\$6.50$ .The probability of landing on an even number is also $\frac{1}{2}$ , but you win $\$0$ for an even number.The expected value for an even number is $0 \times \left(\frac{1}{2}\right)$ .The expected value for an even number is $\$0$ .
Total expected payoff: The expected value for an odd number is $\$6.50$ .The probability of landing on an even number is also $\frac{1}{2}$ , but you win $\$0$ for an even number.The expected value for an even number is $0 \times \left(\frac{1}{2}\right)$ .The expected value for an even number is $\$0$ .To find the total expected payoff, add the expected values for odd and even numbers: $\$6.50 + \$0$ .
Total expected payoff: The expected value for an odd number is $\$6.50$ .The probability of landing on an even number is also $\frac{1}{2}$ , but you win $\$0$ for an even number.The expected value for an even number is $0 \times \left(\frac{1}{2}\right)$ .The expected value for an even number is $\$0$ .To find the total expected payoff, add the expected values for odd and even numbers: $\$6.50 + \$0$ .The total expected payoff is $\$6.50$ .