Question

A 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 10% defective smartphones. Sixty-five percent of the shipments come from factory 2 whose shipments contain 86% defective smartphones. The remainder of the shipments comes from factory 3 whose shipments contain 15% smartphones. The company receives a shipment, but does not know the source. A random sample of 18 smartphones is inspected, and three of the smartphones are found to be defective. What is the probability that this shipment came from factory 1?  

Answer 1

For each factory, we calculate the probability of finding 3 defective smartphones out of a sample of 18 smartphones. This is given by the binomial distribution.

The value of p depends on the factory. So, for factory 1 we have:

Also, we have the fraction of shipment from factory 1

For factory 2, we have:

This is nearly zero.

Also, we have the fraction of shipment from factory 2

For factory 3, we have:

Also, we have the fraction of shipment from factory 3

So, the probability that the sample came from factory 1

Answer: The probability that the sample came from factory 1 is 0.6358