Question

At a raffle, 2000 tickets are sold at $5 each for one prize of $1000, 2 prizes of $500, and three $250. You buy one ticket. What is the expected value

Answer 1

To find the expected value of your ticket in the raffle, you'll calculate the probability of winning each prize, the value of each prize, and then use these to find the expected value.

1. **Total tickets:** 2000
2. **Ticket cost:** $5
3. **Prizes:**
- 1 prize of $1000
- 2 prizes of $500 (totaling $1000)
- 3 prizes of $250 (totaling $750)

Now, calculate the total prize pool:

\[
\text{Total prize pool} = 1000 + 1000 + 750 = 2750
\]

Next, calculate the probability of winning each prize:

- **Probability of winning the $1000 prize:**
\[
P(\text{Win } 1000) = \frac{1}{2000}
\]

- **Probability of winning one of the $500 prizes:**
\[
P(\text{Win } 500) = \frac{2}{2000} = \frac{1}{1000}
\]

- **Probability of winning one of the $250 prizes:**
\[
P(\text{Win } 250) = \frac{3}{2000}
\]

Now, we calculate the expected value by multiplying the value of each prize by the probability of winning that prize and sum them up:

\[
\text{Expected value} = P(\text{Win } 1000) \times 1000 + P(\text{Win } 500) \times 500 + P(\text{Win } 250) \times 250 + P(\text{No win}) \times 0
\]

Where \( P(\text{No win}) \) can be calculated as:

\[
P(\text{No win}) = 1 - P(\text{Win } 1000) - P(\text{Win } 500) - P(\text{Win } 250) = 1 - \frac{1}{2000} - \frac{1}{1000} - \frac{3}{2000}
\]

Calculating \( P(\text{No win}) \):

\[
P(\text{Win } 1000) + P(\text{Win } 500) + P(\text{Win } 250) = \frac{1 + 2 + 3}{2000} = \frac{6}{2000} = \frac{3}{1000}
\]

\[
P(\text{No win}) = 1 - \frac{3}{1000} = \frac{997}{1000}
\]

Now we can calculate the expected value:

\[
\text{Expected value} = \left( \frac{1}{2000} \times 1000 \right) + \left( \frac{2}{2000} \times 500 \right) + \left( \frac{3}{2000} \times 250 \right)
\]

Calculating each term:

\[
\frac{1}{2000} \times 1000 = \frac{1000}{2000} = 0.5
\]

\[
\frac{2}{2000} \times 500 = \frac{1000}{2000} = 0.5
\]

\[
\frac{3}{2000} \times 250 = \frac{750}{2000} = 0.375
\]

Adding these:

\[
\text{Expected winnings} = 0.5 + 0.5 + 0.375 = 1.375
\]

Since you paid $5 for the ticket, the expected value of your net outcome (which is the expected winnings minus the cost of the ticket) is:

\[
\text{Net expected value} = 1.375 - 5 = -3.625
\]

Thus, the expected value of your ticket is \(-3.625\).

**Conclusion:** The expected value is approximately \(-3.63\), indicating that on average, you lose about $3.63 per ticket purchased.