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?

Key Concepts

Bayes' Theorem

Bayes' Theorem provides a method for updating the probability estimate for a hypothesis based on new evidence. In this context, it is used to revise the probability of which supplier the shipment came from once the test results (the number of defectives found in the sample) are observed.

Prior Probability

The prior probability is the initial assessment of the probability of an event before new evidence is taken into account. In the problem, the prior probabilities are based on the proportion of shipments received from each supplier, reflecting initial belief about the shipment's source.

Likelihood

The likelihood is the probability of observing the data under a specific hypothesis. Here, it calculates how probable it is to find exactly one defective item in a sample of twenty, given the defect rate of items from each supplier.

Binomial Distribution

The Binomial Distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials with a constant probability of success. In the problem, it is used to calculate the likelihood of finding a certain number of defectives within the sampled items.

Posterior Probability

Posterior probability is the updated probability of a hypothesis after considering new evidence, calculated using Bayes' Theorem. It reflects the probability that the shipment came from a specific supplier after incorporating the information from the test results.

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