1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 23 24 25 Current learning objective: Mean or Expected Value and Standard Deviation Question 5 Prev Score: 0 of 4 points You buy a lottery ticket to a lottery that costs $$ \$ 15 $$ per ticket. There are only 300 tickets available to be sold in this lottery. In this lottery there are one $$ \$ 500 $$ prize(s), two $$ \$ 100 $$ prizes, and fit prizes. Find your expected gain or loss. (Round to 2 decimal places if needed.) Expected gain or loss $$ \approx \$ $$ Enter your answer Note: Represent a gain with a positive answer and a loss with a negative answer. Submit answer Attempts: 0/1 Need help? We apologize, but your instructor has requested that MathGPT not assist with this assignment. But don't be discouraged, you can do it! Ask Math GPT Al Tutoring is no longer avaliable for this assignment.

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UpStudy AI · Accepted Answer

To find the expected gain or loss from buying a lottery ticket, we need to calculate the expected value based on the probabilities of winning and the associated payouts.

### Known Conditions:
- Cost of a lottery ticket: $$ C = 15 $$
- Total tickets available: $$ N = 300 $$
- Number of prizes:
  - One $$ \$500 $$ prize
  - Two $$ \$100 $$ prizes
  - Five $$ \$0 $$ prizes (assuming no other prizes are mentioned)

### Step 1: Calculate the probabilities of winning each prize.
- Probability of winning the $$ \$500 $$ prize:
  $$
  P(500) = \frac{1}{300}
  $$
  
- Probability of winning one of the $$ \$100 $$ prizes:
  $$
  P(100) = \frac{2}{300} = \frac{1}{150}
  $$

- Probability of winning one of the $$ \$0 $$ prizes:
  $$
  P(0) = \frac{5}{300} = \frac{1}{60}
  $$

### Step 2: Calculate the expected value (EV).
The expected value is calculated as follows:
$$
EV = (P(500) \times 500) + (P(100) \times 100) + (P(0) \times 0) - C
$$

Substituting the probabilities and costs:
$$
EV = \left(\frac{1}{300} \times 500\right) + \left(\frac{2}{300} \times 100\right) + \left(\frac{5}{300} \times 0\right) - 15
$$

### Step 3: Perform the calculations.
Calculating each term:
1. $$ \frac{1}{300} \times 500 = \frac{500}{300} \approx 1.67 $$
2. $$ \frac{2}{300} \times 100 = \frac{200}{300} \approx 0.67 $$
3. $$ \frac{5}{300} \times 0 = 0 $$

Now, summing these values:
$$
EV \approx 1.67 + 0.67 + 0 - 15
$$
$$
EV \approx 2.34 - 15 \approx -12.66
$$

### Final Answer:
The expected gain or loss is approximately:
$$
\text{Expected gain or loss} \approx -12.66
$$

This indicates a loss of approximately $$ \$12.66 $$.
The expected loss is approximately \$12.66.

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