# 4-6. Exercises

**Ex 1.** A company (the producer) supplies microprocessors to a manufacturer (the consumer) of electronic equipment. The microprocessors are supplied in batches of 50. The consumer regards a batch as acceptable provided that there are not more than 5 defective microprocessors in the batch. Rather than test all of the microprocessors in the batch, 10 are selected at random and tested.

1. Find the probability that out of a sample of 10, d = 0, 1, 2, 3, 4, 5 are defective when there are actually 5 defective microprocessors in the batch.
2. Suppose that the consumer will accept the batch provided that not more than m defectives are found in the sample of 10.

       a) Find the probability that the batch is accepted when there are 5 defectives in the batch. ​

       b) Find the probability that the batch is rejected when there are 3 defectives in the batch.

**\[ Solution ]**

1. $$P(X=d) = \frac {*{45}C*{(10-d)} \* \_{5} C *{d}} {*{50} C \_{10} }$$ 
2.  a) $$\Sigma P(X=d) = \Sigma \frac {*{45}C*{(10-d)} \* \_{5} C *{d}} {*{50} C \_{10} }$$ ,    $$m ≤ 5,$$ 

         b) $$d =3,$$   $$P(reject \space batch \space with \space 3 \space defects) =$$  $$1- \Sigma P(X=d) = 1- \Sigma \frac {*{45}C*{10-d} \* \_{5} C *{d}} {*{50} C \_{10} }$$ , $$m ≤ 3$$ 

**Ex 2.** A company buys batches of n components. Before a batch is accepted, m of the components are selected at random from the batch and tested. The batch is rejected if more than d components in the sample are found to be below standard.

(a) Find the probability that a batch which actually contains six below-standard components is rejected when n = 20, m = 5 and d = 1.

(b) Find the probability that a batch which actually contains nine below-standard components is rejected when n = 30, m = 10 and d = 1.

**\[ Solution ]**

![](/files/-Lr7EIQ0PxELjWklCetv)


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://kmis.gitbook.io/statistics/chapter-4.-discrete-random-variables/4-6.-exercises-1.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.