A manufacturing company regularly conducts quality control checks at specifi ed periods on all products manufactured. A new order for 2000 light bulbs is due to be delivered to a national do-it-yourself store. Historically, the manufacturing process has a failure rate of 15% and the sample to be tested consists of four randomly selected light bulbs. From this information estimate the following probabilities: 

(a) Find the probability distribution for 0, 1, 2, 3, and 4 defective light bulbs; 

(b) Calculate the probability that at least three will be defective; 

(c) Determine the mean and variance of the distribution.

This example highlights the case of selecting without replacement from a large population.

The eff ect on the sample space can be considered negligible and therefore we can consider the events as independent. Let the random variable, X, represent the number of defective light bulbs from the random sample. This value of X can take the values: 0 defective from 4 bulbs, or 1 defective from 4 bulbs, or 2 defective from 4 bulbs, or 3 defective from 4 bulbs, or all 4 bulbs defective. This can be written as X = 0, 1, 2, 3, 4. For this example we have p = P(success) = P(defective bulb) = 0.15, q = P(not defective) = 1 – p = 0.85, and n = 4.