Question

A home-builder company receives a shipment of 1,000 lightbulbs every Monday. The lightbulbs are then used immediately in newly finished homes. During the past 4 weeks, the builder counted that 20 of the 4,000 bulbs received were defective. Explain how you arrive to each of your answers to the questions below.

a). Given the information provided, what is a natural estimate for the probability that a given lightbulb is defective?

b). Assuming that different lightbulbs are defective or not independently of each other, give an exact formula for the probability that the shipment arriving next Monday will contain at most one defective lightbulb.

c). Give an approximate formula for the probability that the next shipment will contains exactly k defective lightbulbs.

d). Explain and justify how you would estimate the probability that during the next 10 calendar year the number of shipments containing exactly 5 defective lightbulbs will be greater than 95 and give an approximate value for this probability.

Answer 1

(a) The exact distribution of the number of defective bulbs is a Binomial distribution with proability of success (success being defined as getting a defective bulb) equal to 20 ÷ 4000 = 0.005

(b) The exact formula of Binomial distribution for n = 4000 is

Hence, the required probability of getting at most 1 defective bulb is the sum of above formula for x = 0 and x = 1

(c) The approximate formula is obtained using approximation of Binomial distribution as Poisson distribution, when p is very small and n is very large

Hence, the approximate formula is

(d) We first need to find the probability that a particular shipment contains exactly 5 defective lightbulbs. To keep preciseness, we shall use the Binomial formula

Now the desired event is another Bernoulli trial with 520 repetitions, with probability of success as obtained above and desired success to be at least 95. It can be written as

This computation is impossible by hand. Using a probability calculator, the value comes to be 0.3141