91Ó°ÊÓ

Conditional probability is the likelihood of an event or outcome occurring, based on the occurrence of a previous event. In the context of our problem, we want to determine the probability that a microchip came from supplier \(X\) given that it only fails one test. This probability is represented as \(P(X | \text{Fails one test})\).

To calculate this, we utilize Bayes' Theorem, which provides a way to reverse conditional probabilities. It is expressed as:

• \(P(A | B) = \frac{P(B | A) \times P(A)}{P(B)}\)

Here, \(A\) is the event we are interested in (chip from \(X\)), while \(B\) is the observed event (fails one test).

Bayes’ Theorem helps us update our probability assessment since it assumes prior beliefs (the known defectiveness rates and supplier probabilities) and updates them with new evidence (the chip failing one test). This theorem is a powerful tool in statistics, especially when dealing with probabilities that change based on new information.

In this scenario, conditional probability involves several known probabilities such as the likelihood of purchasing from each supplier and the defect rates from these suppliers. Understanding these can help in reasoning about which supplier a defective chip most likely originated from.

Probability distribution describes how the probabilities are distributed over various outcomes. In the given task, it outlines how the microchips are bought from different firms and helps us understand the likelihood of each firm being the source of a defective chip.

The probability distribution is given as:

• \(P(X) = 0.5\) for supplier \(X\)
• \(P(Y) = 0.3\) for supplier \(Y\)
• \(P(Z) = 0.2\) for supplier \(Z\)

This tells us that half of the chips come from \(X\), while \(30\%\) and \(20\%\) come from \(Y\) and \(Z\) respectively.

This information is crucial when determining the overall probability of any chip being defective, as it forms the basis of the aggregate defect rate calculations across all suppliers. We essentially combine these supplier distribution probabilities with the defect rates to find out how probable it is for a defective chip to be sourced from a specific supplier.

Defective microchips are a crucial aspect of the exercise. Each supplier has a specific rate of defective chips:

• Supplier \(X\): \(2\%\) defective
• Supplier \(Y\): \(4\%\) defective
• Supplier \(Z\): \(4\%\) defective

These percentages tell us the quality control measure of each firm. However, it becomes even more interesting when considering the probability that defective chips escape detection. All defective chips have a \(10\%\) chance of being missed, affecting the probability of detecting defects.

For each supplier, the defective microchips that fail 2 or more tests can be categorized:

• Fails 2+ tests: \(40\%\) for \(X\), \(60\%\) for \(Y\), \(80\%\) for \(Z\)

This affects how likely a chip is to fail only one test. Understanding these defective rates, along with other probabilities, helps in applying the distributions correctly in Bayesian calculations. These calculations allow us to pinpoint the probability of a specific supplier being the source of a failing microchip, given it only fails one test.