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There are two different raffles you can enter.\newlineIn raffle A, one ticket will win a `$720` prize, and the other tickets will win nothing. There are 1,0001,000 in the raffle, each costing `$11`.\newlineRaffle B is for a `$370` prize. Out of 125125 tickets, each costing `$5`, one ticket will win the prize, and the other tickets will win nothing.\newlineWhich raffle is a better deal?\newlineChoices:\newline[A]Raffle A\text{[A]Raffle A}\newline[B]Raffle B\text{[B]Raffle B}

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Q. There are two different raffles you can enter.\newlineIn raffle A, one ticket will win a `$720` prize, and the other tickets will win nothing. There are 1,0001,000 in the raffle, each costing `$11`.\newlineRaffle B is for a `$370` prize. Out of 125125 tickets, each costing `$5`, one ticket will win the prize, and the other tickets will win nothing.\newlineWhich raffle is a better deal?\newlineChoices:\newline[A]Raffle A\text{[A]Raffle A}\newline[B]Raffle B\text{[B]Raffle B}
  1. Calculate Expected Value Raffle A: To determine which raffle is a better deal, we need to calculate the expected value for each raffle. The expected value is calculated by multiplying the prize value by the probability of winning that prize. For Raffle A, the probability of winning is 11000\frac{1}{1000} since there is only one winning ticket out of 10001000.\newlineExpected value for Raffle A = Prize value ×\times Probability of winning\newlineExpected value for Raffle A = $720×11000\$720 \times \frac{1}{1000}
  2. Calculate Cost Raffle A: Now let's calculate the expected value for Raffle A.\newlineExpected value for Raffle A = $720×(1/1000)=$0.72\$720 \times (1/1000) = \$0.72\newlineThis means that for every ticket you buy in Raffle A, you can expect to get back $0.72\$0.72 on average.
  3. Calculate Average Loss/Gain Raffle A: Next, we need to calculate the cost of each ticket in Raffle A to compare it with the expected value.\newlineCost of each ticket in Raffle A = $11\$11\newlineNow we subtract the expected value from the cost to see how much is lost or gained per ticket on average.\newlineAverage loss/gain per ticket in Raffle A = Expected value - Cost\newlineAverage loss/gain per ticket in Raffle A = $0.72\$0.72 - $11\$11
  4. Calculate Expected Value Raffle B: Let's calculate the average loss/gain per ticket in Raffle A.\newlineAverage loss/gain per ticket in Raffle A = \$\(0\).\(72\) - \$\(11\) = -\$\(10\).\(28\)\(\newline\)This means that for every ticket you buy in Raffle A, you can expect to lose \$\(10\).\(28\) on average.
  5. Calculate Cost Raffle B: Now let's calculate the expected value for Raffle B. The probability of winning Raffle B is \(\frac{1}{125}\) since there is only one winning ticket out of \(125\). \(\newline\)Expected value for Raffle B = Prize value \(\times\) Probability of winning \(\newline\)Expected value for Raffle B = \(\$370\) \(\times \left(\frac{1}{125}\right)\)
  6. Calculate Average Loss/Gain Raffle B: Now let's calculate the expected value for Raffle B.\(\newline\)Expected value for Raffle B = \(\$370 \times (1/125) = \$2.96\)\(\newline\)This means that for every ticket you buy in Raffle B, you can expect to get back \(\$2.96\) on average.
  7. Compare Raffle A and Raffle B: Next, we need to calculate the cost of each ticket in Raffle B to compare it with the expected value.\(\newline\)Cost of each ticket in Raffle B = \(\$5\)\(\newline\)Now we subtract the expected value from the cost to see how much is lost or gained per ticket on average.\(\newline\)Average loss/gain per ticket in Raffle B = Expected value - Cost\(\newline\)Average loss/gain per ticket in Raffle B = \(\$2.96\) - \(\$5\)
  8. Compare Raffle A and Raffle B: Next, we need to calculate the cost of each ticket in Raffle B to compare it with the expected value.\(\newline\)Cost of each ticket in Raffle B = \(\$5\)\(\newline\)Now we subtract the expected value from the cost to see how much is lost or gained per ticket on average.\(\newline\)Average loss/gain per ticket in Raffle B = Expected value - Cost\(\newline\)Average loss/gain per ticket in Raffle B = \(\$2.96\) - \(\$5\)Let's calculate the average loss/gain per ticket in Raffle B.\(\newline\)Average loss/gain per ticket in Raffle B = \(\$2.96\) - \(\$5\) = -\(\$2.04\)\(\newline\)This means that for every ticket you buy in Raffle B, you can expect to lose \(\$2.04\) on average.
  9. Compare Raffle A and Raffle B: Next, we need to calculate the cost of each ticket in Raffle B to compare it with the expected value.\(\newline\)Cost of each ticket in Raffle B = \(\$5\)\(\newline\)Now we subtract the expected value from the cost to see how much is lost or gained per ticket on average.\(\newline\)Average loss/gain per ticket in Raffle B = Expected value - Cost\(\newline\)Average loss/gain per ticket in Raffle B = \(\$2.96\) - \(\$5\)Let's calculate the average loss/gain per ticket in Raffle B.\(\newline\)Average loss/gain per ticket in Raffle B = \(\$2.96\) - \(\$5\) = \(-\$2.04\)\(\newline\)This means that for every ticket you buy in Raffle B, you can expect to lose \(\$2.04\) on average.To determine which raffle is a better deal, we compare the average loss/gain per ticket for both raffles. The raffle with the smaller average loss (closer to zero or positive) is the better deal.\(\newline\)Comparing Raffle A and Raffle B:\(\newline\)Raffle A average loss/gain per ticket = \(-\$10.28\)\(\newline\)Raffle B average loss/gain per ticket = \(-\$2.04\)
  10. Compare Raffle A and Raffle B: Next, we need to calculate the cost of each ticket in Raffle B to compare it with the expected value.\(\newline\)Cost of each ticket in Raffle B = \(\$5\)\(\newline\)Now we subtract the expected value from the cost to see how much is lost or gained per ticket on average.\(\newline\)Average loss/gain per ticket in Raffle B = Expected value - Cost\(\newline\)Average loss/gain per ticket in Raffle B = \(\$2.96\) - \(\$5\)Let's calculate the average loss/gain per ticket in Raffle B.\(\newline\)Average loss/gain per ticket in Raffle B = \(\$2.96\) - \(\$5\) = \(-\$2.04\)\(\newline\)This means that for every ticket you buy in Raffle B, you can expect to lose \(\$2.04\) on average.To determine which raffle is a better deal, we compare the average loss/gain per ticket for both raffles. The raffle with the smaller average loss (closer to zero or positive) is the better deal.\(\newline\)Comparing Raffle A and Raffle B:\(\newline\)Raffle A average loss/gain per ticket = \(-\$10.28\)\(\newline\)Raffle B average loss/gain per ticket = \(-\$2.04\)Since the average loss per ticket in Raffle B \(-\$2.04\) is less than the average loss per ticket in Raffle A \(-\$10.28\), Raffle B is the better deal.

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