Question

A manufactured lot of buggy whips has 15 items, of which 6 are defective. A random sample of 3 items is chosen to be inspected. Find the probability (to at least 4 decimal places) that the sample contains exactly 1 defective item: (a) if the sampling is done with replacement (the inspector takes an item and puts it back each time). (b) if the sampling is done without replacement (so the inspector takes a subset of 3 items at once).

Answer 1

To find the probability that a sample of 3 items contains exactly 1 defective item, we can use the binomial distribution. If sampling is done with replacement, the probability is approximately 0.3600. If sampling is done without replacement, the probability is approximately 0.4074.

Step-by-step explanation:

To find the probability that a sample of 3 items contains exactly 1 defective item, we can use the binomial distribution.

(a) If the sampling is done with replacement, the probability of selecting a defective item is 6/15. Let X be the number of defective items in the sample. X can take on the values 0, 1, 2, or 3.

We want to find P(X = 1).

`P(X = 1) = (3 choose 1)(6/15)^1(1 - 6/15)^2`

`= 3(6/15)(9/15)^2`

≈ 0.3600

(b) If the sampling is done without replacement, the probability of selecting a defective item decreases with each selection.

Let X be the number of defective items in the sample. X can take on the values 0, 1, 2, or 3.

We want to find P(X = 1).

P(X = 1) = (6 choose 1)(9 choose 2)/(15 choose 3)

≈ 0.4074