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There are two different raffles you can enter. Raffle A is for a $\$280$ prize. Out of $100$ tickets, each costing $\$15$ , one ticket will win the prize, and the rest will win nothing. In raffle B, one ticket out of $100$ will win a $\$560$ prize, and one ticket will win a $\$490$ prize. The remaining tickets will win nothing. Each ticket costs $\$20$ . Which raffle is a better deal? $\newline$ (A)Raffle A $\newline$ (B)Raffle B

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Choose the better bet

Full solution

Q. There are two different raffles you can enter. Raffle A is for a

\$280

prize. Out of

100

tickets, each costing

\$15

, one ticket will win the prize, and the rest will win nothing. In raffle B, one ticket out of

100

will win a

\$560

prize, and one ticket will win a

\$490

prize. The remaining tickets will win nothing. Each ticket costs

\$20

. Which raffle is a better deal?

\newline

(A)Raffle A

\newline

(B)Raffle B

Calculate Expected Value Raffle A: Calculate the expected value for Raffle A. $\newline$ Expected value = (\text{Probability of winning} \times \text{Prize value}) - (\text{Cost of ticket}) $\newline$ Expected value for Raffle A = $\left(\frac{1}{100} \times \$(280)\right) - \$(15)$ $\newline$ Expected value for Raffle A = $\$(2.80) - \$(15)$ $\newline$ Expected value for Raffle A = $-\$(12.20)$
Calculate Expected Value Raffle B: Calculate the expected value for Raffle B. $\newline$ Expected value = (Probability of winning the first prize $\times$ First prize value) + (Probability of winning the second prize $\times$ Second prize value) - (Cost of ticket) $\newline$ Expected value for Raffle B = $\left(\frac{1}{100} \times \$560\right) + \left(\frac{1}{100} \times \$490\right) - \$20$ $\newline$ Expected value for Raffle B = $\$5.60 + \$4.90 - \$20$ $\newline$ Expected value for Raffle B = $\$10.50 - \$20$ $\newline$ Expected value for Raffle B = $-\$9.50$
Compare Expected Values: Compare the expected values of both raffles to determine which is a better deal. $\newline$ Raffle A has an expected value of $-\$12.20$ and Raffle B has an expected value of $-\$9.50$ . $\newline$ Since the expected value is less negative for Raffle B, it is the better deal.

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