Let's break down each part of this problem.

1. Expected value (to you) of one raffle ticket:

When you buy one raffle ticket, there are two possible outcomes:

• You win: Probability of winning = $\frac{1}{150}$
  • Value if you win = Prize amount - Cost of ticket = $60 - 3 = 57$ dollars.
• You lose: Probability of losing = $\frac{149}{150}$
  • Value if you lose = $-3$ dollars.

The expected value (EV) of one ticket is calculated as: $\text{EV} = \left(\frac{1}{150} \times 57\right) + \left(\frac{149}{150} \times (-3)\right)$ Calculating this: $\text{EV} = \left(\frac{1}{150} \times 57\right) + \left(\frac{149}{150} \times -3\right)$ $\text{EV} \approx 0.38 - 2.98 = -2.60$ So, the expected value to you of one raffle ticket is approximately -$2.60.

2. Expected value (to you) if you purchase 10 raffle tickets:

If you buy 10 tickets, we multiply the expected value of one ticket by 10: $\text{EV for 10 tickets} = 10 \times (-2.60) = -26.00$ So, the expected value to you if you purchase 10 tickets is -$26.00.

3. Expected value (to the PTO) of one raffle ticket:

The expected value to the PTO per ticket is the cost of the ticket minus the expected payout to the winner.

Since the PTO collects $3 per ticket and expects to pay out the equivalent of $0.40 per ticket in prize value (rounded from the previous $0.38), the EV for PTO per ticket is: $\text{EV (to PTO)} = 3 - 0.40 = 2.60$ So, the expected value to the PTO for one ticket is $2.60.

4. Expected total money raised by the PTO if they sell all 150 tickets:

If all 150 tickets are sold, the total amount of money raised is: $150 \times 3 = 450 \text{ dollars}$ The PTO will then pay out $60 to the winner, so the expected net amount is: $450 - 60 = 390 \text{ dollars}$ Thus, if the PTO sells all 150 tickets, they can expect to raise $390 for classroom supplies.

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Let me know if you’d like further details on any part of these calculations!

Related Questions:

1. What is the probability of winning if you buy multiple tickets?
2. How would the expected value change if the prize amount increased?
3. What would the PTO's expected revenue be if they sold fewer than 150 tickets?
4. How does the probability of winning affect the expected value?
5. If the ticket price increased, how would that affect the PTO’s earnings?

Tip:

When calculating expected values, remember to consider both the probability of each outcome and the net gain or loss for each outcome.