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There are two different raffles you can enter. Raffle A has $50$ tickets. Each ticket costs $\$5$ . One ticket will win a $\$190$ prize, and the remaining tickets will win nothing. In raffle B, one ticket out of $1,000$ will win a $\$690$ prize, and twelve tickets will win a $\$650$ prize. The rest will win nothing. Each ticket costs $\$9$ . 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 has

50

tickets. Each ticket costs

\$5

. One ticket will win a

\$190

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

1,000

will win a

\$690

prize, and twelve tickets will win a

\$650

prize. The rest will win nothing. Each ticket costs

\$9

. Which raffle is a better deal?

\newline

(A)Raffle A

\newline

(B)Raffle B

Calculate EVA for Raffle A: Calculate the expected value for Raffle A. $\newline$ Expected value (EVA) = (Probability of winning $\times$ Prize value) + (Probability of not winning $\times$ $0$ ) $\newline$ EVA = ( $\frac{1}{50}$ $\times$ $\$190$ ) + ( $\frac{49}{50}$ $\times$ $\$0$ ) $\newline$ EVA = $\$3.80$ + $\$0$ $\newline$ EVA = $\$3.80$
Subtract cost for expected profit A: Subtract the cost of one ticket from the expected value to find the expected profit for Raffle A. $\newline$ Expected profit (EPA) = Expected value - Cost per ticket $\newline$ $EPA = \$3.80 - \$5$ $\newline$ $EPA = -\$1.20$
Calculate EVB for Raffle B: Calculate the expected value for Raffle B. $\newline$ Expected value (EVB) = (Probability of winning the $\$$ $690$ prize $\times$ Prize value) + (Probability of winning the $\$$ $650$ prize $\times$ Prize value) + (Probability of not winning $\times$ $0$ ) $\newline$ EVB = $\frac{1}{1000} \times \$690$ + $\frac{12}{1000} \times \$650$ + $\frac{987}{1000} \times \$0$ $\newline$ EVB = \$\(0\).\(69\) + \$\(7\).\(80\)\(\newline\)EVB = \$\(8\).\(49\)
Subtract cost for expected profit B: Subtract the cost of one ticket from the expected value to find the expected profit for Raffle B.\(\newline\)Expected profit (EPB) = Expected value - Cost per ticket\(\newline\)\(EPB = \$8.49 - \$9\)\(\newline\)\(EPB = -\$0.51\)
Compare expected profits: Compare the expected profits of Raffle A and Raffle B to determine which is the better deal.\(\newline\)Since \(-\$1.20 < -\$0.51\), Raffle B is the better deal.

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