The PTO is selling raffle tickets to raise money for classroom supplies. A raffle ticket costs $3. There is 1 winning ticket out of the 200 tickets sold. The winner gets a prize worth $66. Round your answers to the nearest cent.

What is the expected value (to you) of one raffle ticket? $


Calculate the expected value (to you) if you purchase 10 raffle tickets. $


What is the expected value (to the PTO) of one raffle ticket? $


If the PTO sells all 200 raffle tickets, how much money can they expect to raise for the classroom supplies? $


To calculate the expected values, we can use the formula for expected value (EV):

\[
EV = (P(win) \times \text{Prize}) + (P(lose) \times \text{Loss})
\]

### 1. Expected Value of One Raffle Ticket

- **Winning Probability (P(win))**: There is 1 winning ticket out of 200.
  \[
  P(win) = \frac{1}{200} = 0.005
  \]

- **Losing Probability (P(lose))**: There are 199 losing tickets.
  \[
  P(lose) = \frac{199}{200} = 0.995
  \]

- **Prize**: $66 (if you win)
- **Loss**: The cost of the ticket, which is $3 (if you lose)

Now, plug these into the EV formula:
\[
EV = \left(\frac{1}{200} \times 66\right) + \left(\frac{199}{200} \times (-3)\right)
\]

Calculating:
\[
EV = \left(0.005 \times 66\right) + \left(0.995 \times -3\right)
\]
\[
EV = 0.33 - 2.985 = -2.655
\]

**Rounded to the nearest cent**:
\[
\text{EV for one ticket} \approx -2.66
\]

### 2. Expected Value for 10 Raffle Tickets

If you buy 10 tickets, your expected value would be 10 times the EV of one ticket:
\[
EV(10) = 10 \times EV(1) = 10 \times (-2.655) = -26.55
\]

**Rounded to the nearest cent**:
\[
\text{EV for 10 tickets} \approx -26.55
\]

### 3. Expected Value to the PTO of One Raffle Ticket

The expected value to the PTO is the income generated minus the expected payout:
- **Income from ticket sale**: $3
- **Expected payout (Prize)**: $ P(win) \times \text{Prize} $
\[
\text{Expected payout} = \frac{1}{200} \times 66 = 0.33
\]

So:
\[
EV_{\text{PTO}} = 3 - 0.33 = 2.67
\]

### 4. Expected Revenue for the PTO if Selling All 200 Tickets

Total revenue is the total income from ticket sales minus the expected payout:
- **Total income**: 
\[
200 \times 3 = 600
\]

- **Total expected payout**: 
\[
\text{Expected payout} = 0.33 \times 200 = 66
\]

So the expected amount raised for classroom supplies is:
\[
600 - 66 = 534
\]

### Summary of Results
1. Expected value for one ticket: **$-2.66**
2. Expected value for 10 tickets: **$-26.55**
3. Expected value to the PTO for one ticket: **$2.67**
4. Expected amount raised for classroom supplies: **$534**

Would you like more details or have any questions? Here are some related questions to consider:
1. How does the expected value change with a different prize amount?
2. What would happen if the number of tickets sold increased?
3. How would the expected value be affected by selling fewer tickets?
4. What strategies could the PTO use to increase ticket sales?
5. How does the concept of expected value apply to other fundraising activities? 

**Tip**: When assessing the value of a gamble or a raffle, always consider both potential winnings and the likelihood of losing your investment.

The PTO is selling raffle tickets to raise money for classroom supplies. A raffle ticket costs $3. There is 1 winning ticket out of the 200 tickets sold. The winner gets a prize worth $66. Round your answers to the nearest cent. What is the expected value (to you) of one raffle ticket? Calculate the expected value (to you) if you purchase 10 raffle tickets. What is the expected value (to the PTO) of one raffle ticket? If the PTO sells all 200 raffle tickets, how much money can they expect to raise for the classroom supplies?

Explore how to calculate the expected value of raffle tickets sold by the PTO to raise money for classroom supplies. With a ticket cost of $3 and a prize worth $66, discover the expected values for one ticket, ten tickets, and the PTO's revenue. Learn important probability concepts and expected value calculations suitable for Grades 9-12.

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