Question

Tickets for a raffle cost $13. There were 753 tickets sold. One ticket will be randomly selected as the winner, and that person wins $1400. For someone who buys a ticket, what is the expected value?

Answer 1

To calculate the expected value for a person who buys a raffle ticket, we need to consider the potential outcomes and their probabilities.

1. **Winning the Raffle**:
- If a person wins, they receive $1400.
- The probability of winning is \( \frac{1}{753} \) (since there is 1 winning ticket out of 753 sold).

2. **Not Winning the Raffle**:
- If a person does not win, they receive nothing.
- The probability of not winning is \( \frac{752}{753} \) (since there are 752 losing tickets).

3. **Net Money Calculation**:
- If the person wins, their net gain will be \( 1400 - 13 = 1387 \) (the prize minus the cost of the ticket).
- If the person loses, their net gain will be \( 0 - 13 = -13 \) (losing the cost of the ticket).

4. **Expected Value Calculation**:
The expected value (EV) can be calculated as follows:

\[
EV = (P(\text{win}) \times \text{net gain if win}) + (P(\text{lose}) \times \text{net gain if lose})
\]

Substituting in the values we derived:

\[
EV = \left( \frac{1}{753} \times 1387 \right) + \left( \frac{752}{753} \times -13 \right)
\]

Calculating each term separately:

- For the winning outcome:
\[
\frac{1}{753} \times 1387 \approx 1.842
\]

- For the losing outcome:
\[
\frac{752}{753} \times -13 \approx -12.934
\]

Now, adding these two results for the overall expected value:

\[
EV \approx 1.842 - 12.934 \approx -11.092
\]

Thus, the expected value for someone who buys a ticket is approximately **-11.09** dollars.

This means that, on average, a ticket buyer can expect to lose about $11.09 from the $13 ticket price.