1.A school fundraiser sells 1,200 raffle tickets. Each ticket costs $2. There is one grand prize worth $100 and 5 smaller prizes worth $20 each. What is the expected gain or loss for buying 1 ticket?
2.Divide.
The correct answer and explanation is:
Problem 1: Expected Gain or Loss for Buying One Ticket
Step 1: Calculate the probability of each outcome
- Total tickets sold: 1,200
- Probability of winning the grand prize = 11200\frac{1}{1200}
- Probability of winning a smaller prize = 51200\frac{5}{1200} for each smaller prize.
- Probability of no prize = 1−11200−51200=119412001 – \frac{1}{1200} – \frac{5}{1200} = \frac{1194}{1200}.
Step 2: Determine the monetary outcomes
- Grand prize: Gain of $100 −2=$98- 2 = \$98.
- Smaller prizes: Gain of $20 −2=$18- 2 = \$18.
- No prize: Loss of $2.
Step 3: Calculate expected value
Expected value (EV) = Sum of (Probability × Outcome)\text{Sum of (Probability × Outcome)}: EV=(11200×98)+(51200×18)+(11941200×−2)EV = \left(\frac{1}{1200} \times 98\right) + \left(\frac{5}{1200} \times 18\right) + \left(\frac{1194}{1200} \times -2\right) EV=0.0817+0.075+(−1.99)=−1.8333EV = 0.0817 + 0.075 + (-1.99) = -1.8333
Expected Loss: $1.83 per ticket.
Explanation
The expected value helps predict the average outcome of an event if repeated many times. Here, buying a ticket involves the risk of losing money since the expected value is negative. While there’s a small chance of winning, the prizes are not large enough compared to the number of tickets sold and the ticket cost. This calculation explains why lotteries and raffles are generally profitable for organizers: participants collectively lose more money than they win, on average.
Problem 2: Division Request Clarification
Please clarify the division problem you’d like solved, as it wasn’t specified. Include numbers or expressions to proceed.