Question

A computer producing factory has only two plants \(T_{1}\) and \(T_{2} .\) Plant \(T_{1}\) produces \(20 \%\) and plant \(T_{2}\) produces \(80 \%\) of the total computers produced. \(7 \%\) of computers produced in the factory turn out to be defective. It is known that \(P\) (computer turns out to be defective given that it is produced in plant \(T_{1}\) ) \(=10 P\) (computer turns out to be defective given that it is produced in plant \(T_{2}\) ), where \(P(E)\) denotes the probability of an event \(E .\) A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant \(T_{2}\) is (A) \(\frac{36}{73}\) (B) \(\frac{47}{79}\) (C) \(\frac{78}{93}\) (D) \(\frac{75}{83}\)

Short answer

\frac{36}{73}

Step-by-step solution

Define Variables

Let the probability a computer is defective from plant T1 be P(D|T1) and from plant T2 be P(D|T2). It is given that P(D|T1) = 10 * P(D|T2). The total probability of a computer being defective is 7%, so P(D) = 0.07.

Set Up Equations

Using the law of total probability: P(D) = P(D|T1) * P(T1) + P(D|T2) * P(T2). Substituting given probabilities: 0.07 = P(D|T1) * 0.20 + P(D|T2) * 0.80, and P(D|T1) = 10 * P(D|T2).

Solve for P(D|T2)

Substitute P(D|T1) in terms of P(D|T2) into the equation from step 2 to find P(D|T2).

Compute P(D|T1)

Use the relation P(D|T1) = 10 * P(D|T2) to calculate P(D|T1) using the value obtained for P(D|T2) from the previous step.

Apply Bayes' Theorem

Bayes' theorem states that P(T2|D') = P(D'|T2) * P(T2) / P(D'), where D' is the event of a computer not being defective. Using the complement rule P(D') = 1 - P(D), find P(D').

Compute P(T2|D')

Use the values from previous steps to find the probability P(T2|D') that the computer was produced in plant T2 given that it is not defective.

Choose the Correct Option

Match the resulting probability P(T2|D') with one of the given options (A), (B), (C), or (D).

Key concepts

Total Probability Theorem

The Total Probability Theorem is a fundamental rule that helps us determine the probability of an event based on several different ways the event can occur. This is especially useful when we deal with processes that have multiple stages or categories, like the manufacturing of computers in different plants.

For example, if a factory has two plants, say, Plant A and Plant B, which produce different quantities of the total output, the total probability of an event E occurring (like producing a defective computer) can be calculated by summing the weighted probabilities of E occurring in each plant individually. This can be formally expressed as:

• \(P(E) = P(E|A)P(A) + P(E|B)P(B)\)

where:

• \(P(E|A)\) is the probability of event E given that it has occurred in Plant A,
• \(P(A)\) is the probability of selecting an item produced in Plant A,
• \(P(E|B)\) is the probability of event E given that it has occurred in Plant B,
• \(P(B)\) is the probability of selecting an item produced in Plant B.

This theorem is essential when we need to consider all possible sources of an item or outcome.

Bayes' Theorem

Bayes' theorem allows us to update our beliefs or probabilities as new information becomes available. When we want to reverse conditional probabilities, which means we need to find the probability of the cause given an outcome, Bayes' theorem is the tool to use.

For instance, if a randomly selected computer from our factory is not defective, and we want to know the probability that it was produced by Plant T2, Bayes' theorem guides us:

• \(P(T2|D') = \frac{P(D'|T2)P(T2)}{P(D')}\)

where:

• \(P(T2|D')\) is the probability that the computer comes from Plant T2 given it's not defective,
• \(P(D'|T2)\) is the probability that a computer is not defective given that it's from Plant T2,
• \(P(T2)\) is the overall probability that a computer comes from Plant T2,
• \(P(D')\) is the probability that a randomly chosen computer is not defective.

Bayes' theorem connects these probabilities and gives us the updated belief based on the new evidence that the computer is functional. This theorem is a powerful statistical tool used not only in quality control but also in various fields like medical diagnosis and machine learning.

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already taken place. It's denoted as \(P(A|B)\), which is the probability of event A occurring given that B has occurred.

In the context of our factory, if we've got two plants, T1 and T2, with different defect rates, and we want to know the probability that a defective computer came from T1 (event A), given that we've already picked a defective computer (event B), we would be looking at the conditional probability \(P(T1|D)\). Conditional probability is pivotal in understanding the relationships between different events and is widely used in risk assessment and predictive analytics. Understanding how to calculate and interpret conditional probability enables us to better assess situations where multiple factors come into play.

Complement Rule

The complement rule is a simple yet powerful concept in probability theory that relates the probability of an event to the probability of its complement. The rule states that the probability of an event not occurring (the complement) is 1 minus the probability of the event occurring.

In mathematical terms:

• \(P(A') = 1 - P(A)\)

where \(P(A')\) is the probability of event A not happening and \(P(A)\) is the probability of event A happening.

When applied to quality control in manufacturing, as with our factory scenario, knowing the probability of a computer being defective (\(P(D)\)), we can easily find the probability of a computer not being defective (\(P(D')\)) using the complement rule. Understanding the complement rule is crucial when dealing with probabilities, as it allows for a complete view of all potential outcomes and is extensively used in risk management.