A smartphone company receives shipments of smartphones from three factories, labelled, 1, 2 and 3. Twenty-five percent of shipments come from factory 1 whose shipments contain 8% defective smartphones. Sixty-five percent of the shipments come from factory 2 whose shipments contain 6% defective smartphones. The remainder of the shipments comes from factory 3 whose shipments contain 4% smartphones. The company receives a shipment, but does not know the source. A random sample of 15 smartphones is inspected, and three of the smartphones are found to be defective. What is the probability that this shipment came from factory 2?

Question

A smartphone company receives shipments of smartphones from three factories, labelled, 1, 2 and 3. Twenty-five percent of shipments come from factory 1 whose shipments contain 8% defective smartphones. Sixty-five percent of the shipments come from factory 2 whose shipments contain 6% defective smartphones. The remainder of the shipments comes from factory 3 whose shipments contain 4% smartphones. The company receives a shipment, but does not know the source. A random sample of 15 smartphones is inspected, and three of the smartphones are found to be defective. What is the probability that this shipment came from factory 2?

Answer 1

To find the probability that the shipment came from factory 2, we can use Bayes' Theorem.

First, let's define the events:
A: The shipment came from factory 1
B: The shipment came from factory 2
C: The shipment came from factory 3
D: The shipment has 3 defective smartphones

We want to find the probability of event B given D, P(B|D).

Bayes' Theorem states:
P(B|D) = (P(D|B) * P(B)) / P(D)

We can break down the formula into three parts:

P(D|B): The probability of finding 3 defective smartphones in a random sample from factory 2. The defective rate is given as 6%, so we want to find the probability of selecting 3 defective smartphones out of 15. This can be calculated using the binomial distribution formula:
P(D|B) = (15 choose 3) * (0.06)^3 * (1 - 0.06)^(15-3)

P(B): The prior probability of a shipment coming from factory 2. Given that 65% of shipments come from factory 2, we can calculate:
P(B) = 0.65

P(D): The probability of finding 3 defective smartphones in the random sample. This can be calculated by considering all possible sources of the shipment and summing their probabilities:
P(D) = P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C)
= (0.08^3 * 0.92^12) * 0.25 + (0.06^3 * 0.94^12) * 0.65 + (0.04^3 * 0.96^12) * 0.10

Now we can apply Bayes' Theorem:
P(B|D) = (P(D|B) * P(B)) / P(D)
= [(15 choose 3) * (0.06)^3 * (1 - 0.06)^(15-3) * 0.65] / [P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C)]

Calculate each component and solve for the probability.