50 screws are picked from 1000 screws with 1%, 5% with major, minor defects, find expected no of defectives.

Question

A lot of 1000 screws contain 1% with major defects and 5% with minor defects. If 50 screws are picked at random and inspected, what are the expected numbers of major and minor defectives?

My attempt:

Probability of major defects = $\frac{1}{100}$

Probability of minor defects = $\frac{5}{100}$

Expected number of major defectives = $50 \times \frac{1}{100}=0.5$

Expected number of minor defectives = $50 \times \frac{5}{100}=2.5$

I am unsure about this because the total number of screws, i.e. $1000$ has not been used here. Is my solution correct?

Answer 1

Re-formulating your question like this, it will be clearer.

The screws produced contain 1% with major defects and 5% with minor defects. If 50 screws are picked at random and inspected out of 1000 screws, what are the expected numbers of major and minor defectives?

Answer 2

In fact we face a https://en.wikipedia.org/wiki/Hypergeometric_distribution with $N=1000$, $K=\text{$1$% of $1000$}=10$ and $n=50$ for the major defects. Hence the expected value is $$n\cdot\frac{K}{N}=50\cdot\frac{10}{1000}.$$ Here's where $1000$ is used.