A lot of 10 electric motors must either be totally rejected or is sold, depending on the outcome of the following procedure: Two motors are chosen at random and inspected. If one or more are defective, the lot is rejected. Otherwise, it is accepted. Suppose that each motor costs $75 and is sold for $100. If the lot contains 1 defective motor, what is the manufacturer's expected profit?  

To calculate the manufacturer's expected profit, we need to consider the probability that the lot will be accepted or rejected based on the inspection procedure.

### Key Information:
1. The lot contains **10 electric motors**.
2. **2 motors are chosen at random** for inspection.
3. If **one or more** of the chosen motors is defective, the lot is rejected.
4. Each motor costs **$75** to produce.
5. The entire lot of 10 motors is sold for **$100 per motor**, so the revenue from selling the lot is **$100 × 10 = $1000**.
6. The lot contains exactly **1 defective motor**.

### Step 1: Compute the probability of rejection and acceptance.

- There are **10 motors in total**, and 1 of them is defective.
- We are selecting 2 motors randomly from this lot.

#### Total number of ways to choose 2 motors from 10:
This is given by the combination formula:

\[
\binom{10}{2} = \frac{10 \times 9}{2} = 45
\]

#### Number of ways to choose 2 good motors:
Since there are 9 good motors and we are choosing 2, the number of ways to select 2 good motors is:

\[
\binom{9}{2} = \frac{9 \times 8}{2} = 36
\]

#### Probability of choosing 2 good motors (lot accepted):
The probability of both motors being good is the ratio of favorable outcomes to total outcomes:

\[
P(\text{both good}) = \frac{\binom{9}{2}}{\binom{10}{2}} = \frac{36}{45} = \frac{4}{5}
\]

Thus, the probability of accepting the lot is $ \frac{4}{5} $.

#### Probability of choosing 1 or more defective motors (lot rejected):
The probability of rejecting the lot is:

\[
P(\text{rejection}) = 1 - P(\text{acceptance}) = 1 - \frac{4}{5} = \frac{1}{5}
\]

### Step 2: Calculate expected profit.

The profit depends on whether the lot is accepted or rejected.

#### Profit if the lot is accepted:
- If accepted, the manufacturer sells the lot for **$1000**.
- The cost of producing 10 motors is **$75 × 10 = $750**.
- Profit if accepted = Revenue − Cost = **$1000 − $750 = $250**.

#### Profit if the lot is rejected:
- If rejected, the manufacturer cannot sell the lot, so the revenue is **$0**.
- However, the production cost of **$750** has still been incurred.
- Profit if rejected = Revenue − Cost = **$0 − $750 = −$750** (a loss).

### Step 3: Compute the expected profit.

The expected profit $ E(\text{profit}) $ is given by:

\[
E(\text{profit}) = P(\text{acceptance}) \times \text{profit if accepted} + P(\text{rejection}) \times \text{profit if rejected}
\]

Substituting the probabilities and profits:

\[
E(\text{profit}) = \frac{4}{5} \times 250 + \frac{1}{5} \times (-750)
\]

\[
E(\text{profit}) = \frac{1000}{5} - \frac{750}{5} = 200 - 150 = 50
\]

### Final Answer:
The manufacturer's expected profit is **$50**.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the expected profit change if the lot contained more than one defective motor?
2. What is the probability of rejecting the lot if two defective motors are present?
3. How does increasing the production cost affect the expected profit?
4. What would the expected profit be if each motor was sold for $150 instead of $100?
5. What is the significance of the number of motors inspected in such a procedure?

**Tip**: Understanding how to calculate probabilities using combinations is crucial when dealing with random selection problems.

A lot of 10 electric motors must either be totally rejected or is sold, depending on the outcome of the following procedure: Two motors are chosen at random and inspected. If one or more are defective, the lot is rejected. Otherwise, it is accepted. Suppose that each motor costs $75 and is sold for $100. If the lot contains 1 defective motor, what is the manufacturer's expected profit?

Discover how to calculate the manufacturer's expected profit when a lot of 10 electric motors is tested for defects. Learn the step-by-step process of probability and expected value, including detailed use of combinatorics and production costs. Suitable for Grades 11-12.

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