Chapter 4 HW - Discrete Random Variable Current score: Submit assignment $$ { }^{+} $$A! Category: Homework (1) 2 (4) 56 8 9 10 11 12 13 14 15 16 17 18 19 20 $$ 21 \quad 22 \quad 23 $$ 28/100 points Current learning objective: Mean or Expected Value and Standard Deviation Question 8 Practice similar questions Prev Next 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 two $$ \$ $$ 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 difficult? Do you know you can practice another version of this question?

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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 of the lottery ticket. The expected value (EV) is calculated using the formula:

$$
EV = \sum (P_i \times V_i)
$$

where $$ P_i $$ is the probability of each outcome and $$ V_i $$ is the value of each outcome.

### Step 1: Identify the outcomes and their probabilities

1. **Winning \$500**: There are 2 prizes of \$500.
   - Probability of winning \$500: 
   $$
   P(\$500) = \frac{2}{1000} = 0.002
   $$
   - Value of winning \$500: 
   $$
   V(\$500) = 500
   $$

2. **Winning \$100**: There are 5 prizes of \$100.
   - Probability of winning \$100: 
   $$
   P(\$100) = \frac{5}{1000} = 0.005
   $$
   - Value of winning \$100: 
   $$
   V(\$100) = 100
   $$

3. **Winning \$25**: There are 20 prizes of \$25.
   - Probability of winning \$25: 
   $$
   P(\$25) = \frac{20}{1000} = 0.02
   $$
   - Value of winning \$25: 
   $$
   V(\$25) = 25
   $$

4. **Not winning anything**: There are 973 tickets that do not win.
   - Probability of not winning: 
   $$
   P(\$0) = \frac{973}{1000} = 0.973
   $$
   - Value of not winning: 
   $$
   V(\$0) = 0
   $$

### Step 2: Calculate the expected value

Now we can calculate the expected value using the probabilities and values identified:

$$
EV = (P(\$500) \times V(\$500)) + (P(\$100) \times V(\$100)) + (P(\$25) \times V(\$25)) + (P(\$0) \times V(\$0))
$$

Substituting the values:

$$
EV = (0.002 \times 500) + (0.005 \times 100) + (0.02 \times 25) + (0.973 \times 0)
$$

Calculating each term:

1. $$ 0.002 \times 500 = 1 $$
2. $$ 0.005 \times 100 = 0.5 $$
3. $$ 0.02 \times 25 = 0.5 $$
4. $$ 0.973 \times 0 = 0 $$

Now, summing these values:

$$
EV = 1 + 0.5 + 0.5 + 0 = 2
$$

### Step 3: Calculate the expected gain or loss

The cost of the lottery ticket is \$15. Therefore, the expected gain or loss is:

$$
\text{Expected Gain or Loss} = EV - \text{Cost of Ticket}
$$

Substituting the values:

$$
\text{Expected Gain or Loss} = 2 - 15 = -13
$$

### Final Answer

The expected gain or loss is:

$$
\text{Expected Gain or Loss} = -\$13.00
$$

Thus, the expected gain or loss is $$-\$13.00$$.
The expected gain or loss is $$-\$13.00$$.

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