For a fundraiser, there is a raffle with 1,000 tickets. One ticket will win a prize,
even tickets will win a prize, nine tickets will win a prize, and the rest will win
othing. If you have a ticket, what is the expected payoff?

Ask by Willis Stephens. in the United States

Jan 23,2025

Question

For a fundraiser, there is a raffle with 1,000 tickets. One ticket will win a prize,
even tickets will win a prize, nine tickets will win a prize, and the rest will win
othing. If you have a ticket, what is the expected payoff?

Ask by Willis Stephens. in the United States

Jan 23,2025

UpStudy AI · Accepted Answer

To find the expected payoff for a ticket in the raffle, we need to calculate the expected value based on the probabilities of winning each prize and the corresponding amounts.

### Known Conditions:
1. Total tickets: $$ N = 1000 $$
2. Prizes:
   - 1 ticket wins $$ \$290 $$
   - 500 tickets (even tickets) win $$ \$230 $$
   - 9 tickets win $$ \$90 $$
   - The rest win nothing.

### Step-by-Step Calculation:

1. **Calculate the probabilities of winning each prize:**
   - Probability of winning $$ \$290 $$:
     $$
     P(290) = \frac{1}{1000}
     $$
   - Probability of winning $$ \$230 $$:
     $$
     P(230) = \frac{500}{1000} = \frac{1}{2}
     $$
   - Probability of winning $$ \$90 $$:
     $$
     P(90) = \frac{9}{1000}
     $$
   - Probability of winning nothing:
     $$
     P(0) = 1 - P(290) - P(230) - P(90) = 1 - \frac{1}{1000} - \frac{500}{1000} - \frac{9}{1000} = \frac{490}{1000}
     $$

2. **Calculate the expected value (EV):**
   The expected value is calculated as follows:
   $$
   EV = (P(290) \times 290) + (P(230) \times 230) + (P(90) \times 90) + (P(0) \times 0)
   $$

Substituting the probabilities and prizes:
   $$
   EV = \left(\frac{1}{1000} \times 290\right) + \left(\frac{500}{1000} \times 230\right) + \left(\frac{9}{1000} \times 90\right) + \left(\frac{490}{1000} \times 0\right)
   $$

3. **Perform the calculations:**
   - For $$ \$290 $$:
     $$
     \frac{1}{1000} \times 290 = 0.290
     $$
   - For $$ \$230 $$:
     $$
     \frac{500}{1000} \times 230 = 115
     $$
   - For $$ \$90 $$:
     $$
     \frac{9}{1000} \times 90 = 0.81
     $$
   - For nothing:
     $$
     \frac{490}{1000} \times 0 = 0
     $$

4. **Sum the expected values:**
   $$
   EV = 0.290 + 115 + 0.81 + 0 = 116.1
   $$

### Final Answer:
The expected payoff for a ticket in the raffle is $$ \$116.10 $$.
The expected payoff for a ticket is \$116.10.

For a fundraiser, there is a raffle with 1,000 tickets. One ticket will win a prize,
even tickets will win a prize, nine tickets will win a prize, and the rest will win
othing. If you have a ticket, what is the expected payoff?

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Probability Questions

For a fundraiser, there is a raffle with 1,000 tickets. One ticket will win a prize, even tickets will win a prize, nine tickets will win a prize, and the rest will win othing. If you have a ticket, what is the expected payoff?

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Probability Questions

For a fundraiser, there is a raffle with 1,000 tickets. One ticket will win a prize,
even tickets will win a prize, nine tickets will win a prize, and the rest will win
othing. If you have a ticket, what is the expected payoff?