Question

At a raffle, 1500 tickets are sold four prizes of $500, $250, $150, and $75. You buy one ticket. What is the expected value?

Answer 1

To calculate the expected value of a raffle ticket, we need to consider the potential winnings and the probability of each outcome.

In this case, you have the following prizes:

- $500 prize: 1 winner
- $250 prize: 1 winner
- $150 prize: 1 winner
- $75 prize: 1 winner
- No prize: 1496 losers (since 1500 tickets are sold and 4 are winners)

Now, let’s calculate the probabilities of each outcome:

- Probability of winning $500: \( \frac{1}{1500} \)
- Probability of winning $250: \( \frac{1}{1500} \)
- Probability of winning $150: \( \frac{1}{1500} \)
- Probability of winning $75: \( \frac{1}{1500} \)
- Probability of winning nothing: \( \frac{1496}{1500} \)

Next, we calculate the expected value (EV) using the formula:
\[
\text{EV} = \sum (\text{Value of the outcome} \times \text{Probability of the outcome})
\]

Substituting in the values:

\[
\text{EV} = (500 \times \frac{1}{1500}) + (250 \times \frac{1}{1500}) + (150 \times \frac{1}{1500}) + (75 \times \frac{1}{1500}) + (0 \times \frac{1496}{1500})
\]

Calculating each term:

1. For $500 prize:
\[
500 \times \frac{1}{1500} = \frac{500}{1500} = \frac{1}{3} \approx 0.3333
\]

2. For $250 prize:
\[
250 \times \frac{1}{1500} = \frac{250}{1500} = \frac{1}{6} \approx 0.1667
\]

3. For $150 prize:
\[
150 \times \frac{1}{1500} = \frac{150}{1500} = \frac{1}{10} = 0.1
\]

4. For $75 prize:
\[
75 \times \frac{1}{1500} = \frac{75}{1500} = \frac{1}{20} = 0.05
\]

5. For no prize:
\[
0 \times \frac{1496}{1500} = 0
\]

Now sum all these values to get the expected value:

\[
\text{EV} = 0.3333 + 0.1667 + 0.1 + 0.05 + 0
\]
\[
\text{EV} = 0.3333 + 0.1667 + 0.1 + 0.05 = 0.65
\]

Thus, the expected value of your $1 ticket is approximately **$0.65**.