A hospital receives 1 / 5 of its flu vaccine shipments from Company X and the remainder of its s

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A hospital receives 1 / 5 of its flu vaccine shipments from...

A hospital receives $1 / 5$ of its flu vaccine shipments from Company $\mathrm{X}$ and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials. For Company X's shipments, $10 \%$ of the vials are ineffective. For every other company, $2 \%$ of the vials are ineffective. The hospital tests 30 randomly selected vials from a shipment and finds that one vial is ineffective. What is the probability that this shipment came from Company X? Choose one of the following. (Hint: Find the probability that one out of 30 vials is ineffective, given that the shipment came from Company $\mathrm{X}$ and that the shipment came from other companies. Then use Bayes' theorem.) a. 0.10 b. 0.14 c. 0.37 d. 0.63 e. 0.86

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Key Concepts

Bayes' Theorem

Bayes' Theorem is a rule that allows one to update the probability estimate for a hypothesis as more evidence becomes available. It combines the prior probability, which is the initial degree of belief in the hypothesis, with the likelihood of the observed evidence under that hypothesis, resulting in a posterior probability that reflects the updated belief.

Binomial Distribution

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. In contexts where each trial has only two outcomes (such as effective versus ineffective vaccine vials), the binomial distribution is used to calculate the probability of observing a specific number of successes.

Law of Total Probability

The Law of Total Probability is a fundamental rule that breaks down the probability of an event into several disjoint cases or conditional probabilities. It is used to compute the overall probability of an event by summing the probabilities of that event occurring under each scenario, each weighted by the likelihood of that scenario.

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