JEE Mathematics 2023 Question with Solution

Given: Machines A, B, and C manufacture $20\%$, $30\%$, and $50\%$ of the bolts respectively. Their defective rates are $3\%$, $4\%$, and $2\%$ respectively.

Find: The probability that a bolt was manufactured by $C$ given that it is defective.

Use Bayes' Theorem:

First compute the total probability of a defective bolt:

Substituting the given values from the question:

Now apply Bayes' Theorem:

Therefore, the probability that the defective bolt was manufactured by $C$ is $\frac{5}{14}$. Hence, the correct option is A.

The solution applies Bayes' Theorem, which is the correct method, but the numerical substitutions shown there do not match the question statement.

From the question, the correct data are:

• $P(A)=0.20$
• $P(B)=0.30$
• $P(C)=0.50$
• $P(D\mid A)=0.03$
• $P(D\mid B)=0.04$
• $P(D\mid C)=0.02$

The solution instead uses:

• $P(A)=0.50$, $P(B)=0.30$, $P(C)=0.20$
• $P(D\mid A)=0.04$, $P(D\mid B)=0.03$, $P(D\mid C)=0.02$ which swaps the values and leads to $\frac{4}{33}$, a result that does not correspond to the given question or the marked correct answer.

Using the question data exactly as given yields:

So the marked answer A is consistent with the question, while the page's worked values are not.