A company receives large shipments of parts from two...

A company receives large shipments of parts from two sources. Seventy percent of the shipments come from a supplier whose shipments typically contain $10 \%$ defectives, while the remainder are from a supplier whose shipments typically contain $20 \%$ defectives. A manager receives a shipment but does not know the source. A random sample of 20 items from this shipment is tested, and 1 of the parts is found to be defective. What is the probability that this shipment came from the more reliable supplier? (Hint: Use Bayes' theorem.)

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Key Concepts

Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability theory that describes how to update the probabilities of hypotheses when given evidence. In problems like this one, it is used to determine the probability that a shipment came from a particular supplier after observing the outcome of part sampling, by incorporating both the prior information and the likelihood of the observed evidence.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. This concept is critical in updating beliefs about which supplier the shipment came from after testing a sample of items, as it allows one to calculate the probability of the observed sample results conditioned on the known defect rates of the suppliers.

Binomial Distribution

The binomial distribution is used to model the number of successes in a fixed number of independent trials, where each trial has the same probability of success. In the context of this problem, it is used to calculate the likelihood of obtaining a specific number of defective items in the tested sample, given the known defect rates of the different suppliers.

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