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# 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 ]**

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr7DtkzUuyir-qtEZGo%2F-Lr7EIQ0PxELjWklCetv%2Fimage%20\(56\).png?alt=media\&token=9eeaf90d-9248-4aab-913c-b4b842b1173b)