Suppose you are at the Fair and are offered the opportunity to spin a spinner like the one at right and win the dollar amount of the number you land on. The catch is that it costs \( \$ 15 \) to spin the spinner. Would you spin?

To determine whether it is worth spinning the spinner, we need to calculate the expected value of the game. The expected value is calculated by multiplying each possible outcome by its probability and then summing up these products. Let's denote: - \( X \) as the random variable representing the dollar amount won by spinning the spinner. - \( P(X=x) \) as the probability of landing on the number \( x \). - \( E(X) \) as the expected value of the game. Given that the spinner has numbers from 1 to 6, we can assume that each number has an equal probability of \( \frac{1}{6} \) of being landed on. The cost of spinning the spinner is $15. The expected value of the game can be calculated as follows: \[ E(X) = \sum_{x=1}^{6} x \cdot P(X=x) - 15 \] Let's calculate the expected value to determine if it is worth spinning the spinner. Calculate the value by following steps: - step0: Calculate: \(\frac{1}{6}\times 1+\frac{1}{6}\times 2+\frac{1}{6}\times 3+\frac{1}{6}\times 4+\frac{1}{6}\times 5+\frac{1}{6}\times 6-15\) - step1: Multiply: \(\frac{1}{6}+\frac{1}{6}\times 2+\frac{1}{6}\times 3+\frac{1}{6}\times 4+\frac{1}{6}\times 5+\frac{1}{6}\times 6-15\) - step2: Reduce the numbers: \(\frac{1}{6}+\frac{1}{3}+\frac{1}{6}\times 3+\frac{1}{6}\times 4+\frac{1}{6}\times 5+\frac{1}{6}\times 6-15\) - step3: Reduce the numbers: \(\frac{1}{6}+\frac{1}{3}+\frac{1}{2}+\frac{1}{6}\times 4+\frac{1}{6}\times 5+\frac{1}{6}\times 6-15\) - step4: Multiply the numbers: \(\frac{1}{6}+\frac{1}{3}+\frac{1}{2}+\frac{2}{3}+\frac{1}{6}\times 5+\frac{1}{6}\times 6-15\) - step5: Multiply: \(\frac{1}{6}+\frac{1}{3}+\frac{1}{2}+\frac{2}{3}+\frac{5}{6}+\frac{1}{6}\times 6-15\) - step6: Reduce the numbers: \(\frac{1}{6}+\frac{1}{3}+\frac{1}{2}+\frac{2}{3}+\frac{5}{6}+1-15\) - step7: Subtract the numbers: \(\frac{1}{6}+\frac{1}{3}+\frac{1}{2}+\frac{2}{3}+\frac{5}{6}-14\) - step8: Reduce fractions to a common denominator: \(\frac{1}{6}+\frac{2}{3\times 2}+\frac{3}{2\times 3}+\frac{2\times 2}{3\times 2}+\frac{5}{6}-\frac{14\times 6}{6}\) - step9: Calculate: \(\frac{1}{6}+\frac{2}{6}+\frac{3}{6}+\frac{4}{6}+\frac{5}{6}-\frac{84}{6}\) - step10: Transform the expression: \(\frac{1+2+3+4+5-84}{6}\) - step11: Calculate: \(\frac{-69}{6}\) - step12: Calculate: \(-\frac{23}{2}\) The expected value of the game is $-11.5. This means that on average, you would lose $11.50 for each spin of the spinner. Since the expected value is negative, it is not worth spinning the spinner as you would, on average, lose money.