please solve a and b

Question

A manufacturer produces x% of defective light bulbs on the average. Each item is inspected before being put on the market. Items that are classified as good are put on the market and those classified as defective are scrapped. In the past, the inspectors misclassify an item 5% of the time. After inspecting all the light bulbs, it is found that 77% of items are classified as good and put on the market.

(a) Find the percentage of light bulbs that are actually defective.

(b) A bulb is selected at random from those that are put on the market. Find the probability that the item is actually good

Answer 1

(a) To find the percentage of light bulbs that are actually defective, we need to use Bayes' theorem, which relates the conditional probabilities of two events. Let D denote the event that a light bulb is defective, and G denote the event that a light bulb is classified as good. Then:

P(D|G) = P(G|D)*P(D) / P(G)

where P(G|D) is the probability that a defective light bulb is classified as good (i.e., not scrapped), P(D) is the overall percentage of defective light bulbs, and P(G) is the probability that a light bulb is classified as good (i.e., put on the market).

We know that P(G) = 0.77, and we can compute P(G|D) as 1 - 0.05 = 0.95, since the inspectors misclassify a defective light bulb with probability 0.05. Finally, we are given that x% of light bulbs are defective on average, so P(D) = x/100. Substituting these values into Bayes' theorem, we get:

P(D|G) = 0.95*(x/100) / 0.77

Simplifying this expression, we get:

P(D|G) = 1.23x/100

Therefore, the percentage of light bulbs that are actually defective is 1.23 times the average percentage of defective light bulbs.

(b) To find the probability that a randomly selected light bulb from those put on the market is actually good, we need to use the complementary probability:

P(good) = 1 - P(defective)

We have already computed P(defective) in part (a) as 1.23x/100. Therefore:

P(good) = 1 - 1.23x/100

Substituting x = 5% (since the problem does not give a specific value for x), we get:

P(good) = 1 - 1.23(5)/100 = 0.9385

Therefore, the probability that a randomly selected light bulb from those put on the market is actually good is 0.9385.