In a manufacturing plant, machine A produces 10 % of a...

In a manufacturing plant, machine A produces $10 \%$ of a certain product, machine $\mathrm{B}$ produces $40 \%$ of this product, and machine C produces $50 \%$ of this product. Five percent of machine A products are defective, $12 \%$ of machine B products are defective, and $8 \%$ of machine C products are defective. The company inspector has just sampled a product from this plant and has found it to be defective. Determine the revised probabilities that the sampled product was produced by machine $\mathrm{A}$, machine $\mathrm{B}$, or machine $\mathrm{C}$.

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Key Concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. This concept is essential in analyzing events where outcomes depend on additional known conditions or circumstances.

Bayes' Theorem

Bayes' Theorem offers a method to update the probability of a hypothesis as new evidence becomes available. It connects prior probability with the likelihood of observed evidence to compute the revised or posterior probability.

Law of Total Probability

The Law of Total Probability is used to determine the overall probability of an event by summing the probabilities of the event occurring under different, mutually exclusive conditions. This law is pivotal in decomposing complex probability problems into manageable segments.

Defect Rate Analysis

Defect rate analysis deals with calculating the probability of defects in a production process. This involves considering the different defect rates from various sources and the proportion of total output each contributes, which is crucial for quality control and decision-making in manufacturing.

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