Question
At a raffle, 1500 tickets are sold at $2 each for four prizes of $500, $250, $150, and $75. You buy one ticket. What is the expected value of your gain (loss)?
Answers
GPT-4o mini
To calculate the expected value of your gain (or loss) from buying one ticket, we need to consider the probability of winning each prize and the potential outcomes.
1. **Total number of tickets sold:** 1500
2. **Price per ticket:** $2
3. **Prizes:**
- First prize: $500
- Second prize: $250
- Third prize: $150
- Fourth prize: $75
Now, let's calculate the probabilities and outcomes.
### Probability of Winning Each Prize
- **Probability of winning the $500 prize**: \( \frac{1}{1500} \)
- **Probability of winning the $250 prize**: \( \frac{1}{1500} \)
- **Probability of winning the $150 prize**: \( \frac{1}{1500} \)
- **Probability of winning the $75 prize**: \( \frac{1}{1500} \)
- **Probability of winning nothing**: \( \frac{1496}{1500} \)
### Gains from Winning and Losses
- Gain if you win $500: \( 500 - 2 = 498 \) (since you paid $2 for the ticket)
- Gain if you win $250: \( 250 - 2 = 248 \)
- Gain if you win $150: \( 150 - 2 = 148 \)
- Gain if you win $75: \( 75 - 2 = 73 \)
- Loss if you win nothing: \( -2 \)
### Expected Value Calculation
Now we can calculate the expected value (EV):
\[
\text{EV} = P(\text{win } 500) \cdot \text{gain from } 500 + P(\text{win } 250) \cdot \text{gain from } 250 + P(\text{win } 150) \cdot \text{gain from } 150 + P(\text{win } 75) \cdot \text{gain from } 75 + P(\text{win nothing}) \cdot \text{loss}
\]
Substituting the values:
\[
\text{EV} = \left( \frac{1}{1500} \cdot 498 \right) + \left( \frac{1}{1500} \cdot 248 \right) + \left( \frac{1}{1500} \cdot 148 \right) + \left( \frac{1}{1500} \cdot 73 \right) + \left( \frac{1496}{1500} \cdot (-2) \right)
\]
Calculating each term:
1. First prize contribution: \( \frac{498}{1500} \)
2. Second prize contribution: \( \frac{248}{1500} \)
3. Third prize contribution: \( \frac{148}{1500} \)
4. Fourth prize contribution: \( \frac{73}{1500} \)
5. Loss from winning nothing: \( \frac{1496 \cdot (-2)}{1500} = \frac{-2992}{1500} \)
Now, summing all these contributions:
\[
\text{EV} = \frac{498 + 248 + 148 + 73 - 2992}{1500}
\]
Calculating the numerator:
\[
498 + 248 + 148 + 73 = 967
\]
\[
967 - 2992 = -2025
\]
Thus,
\[
\text{EV} = \frac{-2025}{1500} \approx -1.35
\]
### Conclusion
The expected value of your gain (loss) when buying one ticket is approximately **-$1.35**. This means that, on average, you can expect to lose about $1.35 for each ticket purchased in this raffle.
1. **Total number of tickets sold:** 1500
2. **Price per ticket:** $2
3. **Prizes:**
- First prize: $500
- Second prize: $250
- Third prize: $150
- Fourth prize: $75
Now, let's calculate the probabilities and outcomes.
### Probability of Winning Each Prize
- **Probability of winning the $500 prize**: \( \frac{1}{1500} \)
- **Probability of winning the $250 prize**: \( \frac{1}{1500} \)
- **Probability of winning the $150 prize**: \( \frac{1}{1500} \)
- **Probability of winning the $75 prize**: \( \frac{1}{1500} \)
- **Probability of winning nothing**: \( \frac{1496}{1500} \)
### Gains from Winning and Losses
- Gain if you win $500: \( 500 - 2 = 498 \) (since you paid $2 for the ticket)
- Gain if you win $250: \( 250 - 2 = 248 \)
- Gain if you win $150: \( 150 - 2 = 148 \)
- Gain if you win $75: \( 75 - 2 = 73 \)
- Loss if you win nothing: \( -2 \)
### Expected Value Calculation
Now we can calculate the expected value (EV):
\[
\text{EV} = P(\text{win } 500) \cdot \text{gain from } 500 + P(\text{win } 250) \cdot \text{gain from } 250 + P(\text{win } 150) \cdot \text{gain from } 150 + P(\text{win } 75) \cdot \text{gain from } 75 + P(\text{win nothing}) \cdot \text{loss}
\]
Substituting the values:
\[
\text{EV} = \left( \frac{1}{1500} \cdot 498 \right) + \left( \frac{1}{1500} \cdot 248 \right) + \left( \frac{1}{1500} \cdot 148 \right) + \left( \frac{1}{1500} \cdot 73 \right) + \left( \frac{1496}{1500} \cdot (-2) \right)
\]
Calculating each term:
1. First prize contribution: \( \frac{498}{1500} \)
2. Second prize contribution: \( \frac{248}{1500} \)
3. Third prize contribution: \( \frac{148}{1500} \)
4. Fourth prize contribution: \( \frac{73}{1500} \)
5. Loss from winning nothing: \( \frac{1496 \cdot (-2)}{1500} = \frac{-2992}{1500} \)
Now, summing all these contributions:
\[
\text{EV} = \frac{498 + 248 + 148 + 73 - 2992}{1500}
\]
Calculating the numerator:
\[
498 + 248 + 148 + 73 = 967
\]
\[
967 - 2992 = -2025
\]
Thus,
\[
\text{EV} = \frac{-2025}{1500} \approx -1.35
\]
### Conclusion
The expected value of your gain (loss) when buying one ticket is approximately **-$1.35**. This means that, on average, you can expect to lose about $1.35 for each ticket purchased in this raffle.