Q.An item is manufactured by three machines $A$, $B$ and $C$. Out of the total number of items manufactured during a specified period, $50\%$ are manufactured on $A$, $30\%$ on $B$ and $20\%$ on $C$. $2\%$ of the items produced on $A$ and $2\%$ of items produced on $B$ are defective, and $3\%$ of those produced on $C$ are defective. All the items are stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine $A$?

Concept: Conditional Probability (Bayes' Theorem)

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

Step 1 – Write the given probabilities

$P(A) = 0.50$, $P(B) = 0.30$, $P(C) = 0.20$

$P(D \mid A) = 0.02$, $P(D \mid B) = 0.02$, $P(D \mid C) = 0.03$

Step 2 – Find total probability of defect

$P(D) = P(A)P(D \mid A) + P(B)P(D \mid B) + P(C)P(D \mid C)$

$= (0.50)(0.02) + (0.30)(0.02) + (0.20)(0.03)$

$= 0.01 + 0.006 + 0.006 = 0.022$ …

Why Bayes’ theorem fits here

We are told the overall production shares of three machines, and the defect rates within each machine’s output. Then we pick one item, see it is defective, and need the reverse probability: “given that it is defective, what is the chance it came from machine $A$?”

This is the classic setup for Bayes’ theorem — we have prior probabilities (the production shares) and likelihoods (the defect rates), and we want the posterior probability after observing the defect.

Step-by-step solution

1. Define the events clearly

Let $A$, $B$, $C$ denote the events that the item was manufactured on machine $A$, $B$, or $C$ respectively.

Let $D$ denote the event that the item is defective.

2. Write down the given probabilities

• Prior probabilities (production shares):

  $P(A) = 0.50$, $P(B) = 0.30$, $P(C) = 0.20$

• Conditional probabilities (defect rates given the machine):

  $P(D \mid A) = 0.02$, $P(D \mid B) = 0.02$, $P(D \mid C) = 0.03$

3. Find the total probability of a defective item

By the law of total probability:

Substitute:

So $2.2\%$ of all items are defective.

4. Apply Bayes’ theorem for $P(A \mid D)$

Bayes’ theorem says:

Plug in the numbers:

Simplify the fraction: …

Method: Bayes' theorem for "which machine made the defective item"

Use this when items come from several machines with known production shares and defect rates, and you must find the probability a defective item came from a particular machine.

Steps

Step 1: Priors and likelihoods

Priors are the production shares $P(A),P(B),P(C)$; likelihoods are the per-machine defect rates $P(D\mid A),\dots$

Step 2: Total probability of a defect

$P(D)=P(A)P(D\mid A)+P(B)P(D\mid B)+P(C)P(D\mid C).$ …

Common Mistakes

Mistake 1: Using the sum of defect rates as the denominator

Why it's wrong: $0.02+0.02+0.03$ ignores how much each machine produces; the denominator must be production-weighted. Correct approach: weight each defect rate by its machine's share.

Mistake 2: Swapping the conditional direction

Why it's wrong: the defect rate $P(D\mid A)$ is not the wanted $P(A\mid D)$. Correct approach: apply Bayes to reverse it. …