A lot containing 7 components is sampled by a quality inspector, the lot contains 4 good components and 3 defective components. A sample of 3 is taken by the inspector. Find the expected value of the number of good components in this sample.

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To find the expected value of the number of good components in the sample, we can treat this as a **hypergeometric distribution** problem.

### Problem Breakdown:
- The total number of components in the lot: \( N = 7 \)
- Number of good components: \( G = 4 \)
- Number of defective components: \( D = 3 \)
- Sample size: \( n = 3 \)

The hypergeometric distribution gives us the probability of \( k \) good components in a sample of size \( n \) drawn from a population of \( N \) objects, \( G \) of which are good.

The expected value for the number of good components in a hypergeometric distribution is given by the formula:

\[
E(X) = n \times \frac{G}{N}
\]

Where:
- \( n = 3 \) is the sample size,
- \( G = 4 \) is the number of good components,
- \( N = 7 \) is the total number of components.

### Calculating the Expected Value:
\[
E(X) = 3 \times \frac{4}{7} = \frac{12}{7} \approx 1.71
\]

Thus, the expected value of the number of good components in the sample is approximately **1.71**.

Would you like more details or have any questions?

Here are some related questions:
1. What is the probability of drawing exactly 2 good components?
2. How does the expected value change if the lot size increases?
3. How does replacing components (with or without) affect the hypergeometric distribution?
4. Can the expected value ever exceed the sample size?
5. What is the variance of this hypergeometric distribution?

**Tip**: The hypergeometric distribution is used when sampling without replacement, as opposed to the binomial distribution.

A lot containing 7 components is sampled by a quality inspector, the lot contains 4 good components and 3 defective components. A sample of 3 is taken by the inspector. Find the expected value of the number of good components in this sample.

Learn how to calculate the expected value of good components in a sample using hypergeometric distribution. This problem involves finding the expected value for a sample of 3 taken from a lot of 7 components, with 4 good and 3 defective components. Suitable for Grades 9-12.

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