Q.In a factory which manufactures bolts, machines A, B and C manufacture respectively 25%, 35% and 40% of the bolts. Of their outputs, 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B?

Concept: Conditional Probability (Bayes' Theorem)

We want $P(B \mid D)$, the probability the bolt came from machine B given it is defective.

Step 1 – Prior probabilities

$P(A) = 0.25$, $P(B) = 0.35$, $P(C) = 0.40$.

Step 2 – Likelihoods (defect rates)

$P(D \mid A) = 0.05$, $P(D \mid B) = 0.04$, $P(D \mid C) = 0.02$.

Step 3 – Total probability of a defective bolt

$P(D) = 0.25 \times 0.05 + 0.35 \times 0.04 + 0.40 \times 0.02$

$= 0.0125 + 0.0140 + 0.0080 = 0.0345$.

Step 4 – Apply Bayes’ Theorem …

Why Bayes’ theorem?

We are asked: Given that a bolt is defective, what is the chance it came from machine B? This is a reverse probability — we know the chance of a defect given the machine, but we want the chance of the machine given a defect. That’s exactly what Bayes’ theorem handles.

The key insight: the total probability of a defective bolt is a weighted average of the defect rates of the three machines, weighted by their production shares. Then, the share of that total that comes from machine B is the answer.

Step-by-step solution

1. Define events clearly

Let:

• $A$, $B$, $C$ = event that a randomly chosen bolt is made by machine A, B, or C respectively.
• $D$ = event that the bolt is defective.

We are given:

• $P(A) = 0.25$, $P(B) = 0.35$, $P(C) = 0.40$
• $P(D \mid A) = 0.05$, $P(D \mid B) = 0.04$, $P(D \mid C) = 0.02$

We want $P(B \mid D)$.

2. Find the total probability of a defective bolt

By the law of total probability:

Substitute:

Compute each term:

• $0.25 \times 0.05 = 0.0125$
• $0.35 \times 0.04 = 0.0140$
• $0.40 \times 0.02 = 0.0080$

Sum:

So 3.45% of all bolts are defective.

3. Apply Bayes’ theorem

Bayes’ theorem for $P(B \mid D)$:

Plug in:

4. Simplify the fraction …

Method: Bayes' Theorem (finding the cause from the observed outcome)

Use this when items come from several sources and you observe a property (here, a defect) and want the probability it came from a particular source. You are given the defect rate given each machine, but want the machine given a defect — a reversed conditional.

Steps

Step 1: Set up the partition of possible causes.

List the mutually exclusive, exhaustive hypotheses (here: the bolt was made by machine A, B or C) and write their prior probabilities $P(E_i)$ (these must sum to $1$).

Step 2: Write each likelihood.

For each cause, state $P(A\mid E_i)$ — the probability of the observed outcome (here: the bolt is defective) under that cause.

Step 3: Get the total probability of the outcome (the denominator).

By the law of total probability,

Step 4: Apply Bayes' theorem for the cause you want.

$P(E_k\mid A)=\frac{P(E_k)\,P(A\mid E_k)}{\sum_i P(E_i)\,P(A\mid E_i)}.$ …

Common Mistakes

Mistake 1: Comparing only the defect rates.

Why it's wrong: using $\tfrac{4}{5+4+2}=\tfrac{4}{11}$ ignores how many bolts each machine makes. Correct approach: weight each defect rate by the machine's production share.

Mistake 2: Reporting $P(\text{defective}\mid B)=0.04$ as the answer.

Why it's wrong: that is the likelihood, not the required $P(B\mid\text{defective})$. Correct approach: apply Bayes' theorem to reverse the conditioning. …