My Courses CRN 40938 Chapter 4 HW - Discrete Random Variable Current score: 28/100 points *À Category: Homework $$ \begin{array}{lllllllllllllllllllll}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 \ 22 & 23\end{array} $$ Current learning objective: Mean or Expected Value and Standard Deviation Question 8 Practice similar questions Prev Questions Practice similar questions Score: 0 of 3 points You buy a lottery ticket to a lottery that costs $$ \$ 15 $$ per ticket. There are only 1000 tickets available to be sold in this lottery. In this lottery there are 500 prize(s), five $$ \$ 100 $$ prizes, and twenty $$ \$ 25 $$ prizes. Find your expected gain or loss. (Round to 2 decimal places if needed.) Expected gain or loss $$ =\$ $$ Enter your answer Note: Represent a gain with a positive answer and a loss with a negative answer. Submit answer Attempts: 0/10 Find this question difficut? Do you know you can practice another version of this question?

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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. Let's break this down step by step.

### Step 1: Identify the known values
- Cost of a lottery ticket: $$ C = 15 $$
- Total number of tickets: $$ N = 1000 $$
- Number of winning tickets: $$ W = 500 $$ (includes both types of prizes)
- Number of $$ \$100 $$ prizes: $$ P_{100} = 5 $$
- Number of $$ \$25 $$ prizes: $$ P_{25} = 20 $$

### Step 2: Calculate the probabilities
1. **Probability of winning a $$ \$100 $$ prize:**
   $$
   P_{100} = \frac{5}{1000} = 0.005
   $$

2. **Probability of winning a $$ \$25 $$ prize:**
   $$
   P_{25} = \frac{20}{1000} = 0.02
   $$

3. **Probability of winning nothing:**
   $$
   P_{0} = 1 - P_{100} - P_{25} = 1 - 0.005 - 0.02 = 0.975
   $$

### Step 3: Calculate the expected value
The expected value $$ E $$ can be calculated using the formula:
$$
E = (P_{100} \times 100) + (P_{25} \times 25) + (P_{0} \times 0)
$$

Substituting the probabilities:
$$
E = (0.005 \times 100) + (0.02 \times 25) + (0.975 \times 0)
$$

### Step 4: Perform the calculations
Let's calculate $$ E $$:
1. Calculate $$ 0.005 \times 100 $$:
   $$
   0.005 \times 100 = 0.5
   $$

2. Calculate $$ 0.02 \times 25 $$:
   $$
   0.02 \times 25 = 0.5
   $$

3. Combine these results:
   $$
   E = 0.5 + 0.5 + 0 = 1.0
   $$

### Step 5: Calculate the expected gain or loss
The expected gain or loss is calculated by subtracting the cost of the ticket from the expected value:
$$
\text{Expected Gain or Loss} = E - C
$$
Substituting the values:
$$
\text{Expected Gain or Loss} = 1.0 - 15
$$

### Step 6: Final calculation
$$
\text{Expected Gain or Loss} = -14.0
$$

### Conclusion
The expected gain or loss from buying a lottery ticket is:
$$
\text{Expected Gain or Loss} = -\$14.00
$$

Thus, you would enter:
$$
\text{Expected gain or loss} = -14.00
$$
The expected gain or loss from buying a lottery ticket is $$-\$14.00$$.

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