Bayes' Theorem

Question

Two factories supply identical bolts in the ratio 60 to 40, with defect rates 2 percent and 5 percent respectively; a randomly chosen defective bolt is examined, so what is the probability it came from the first factory?

RankGuru · Accepted Answer

Bayes' theorem inverts conditional probabilities, expressing P(cause | effect) in terms of prior probabilities and likelihoods through P(Aᵢ | D) = P(Aᵢ)P(D | Aᵢ) / ΣP(Aⱼ)P(D | Aⱼ). This is the classic JEE Advanced source-identification problem. Let factory 1 supply 0.6 of bolts with defect rate 0.02 and factory 2 supply 0.4 with defect rate 0.05. The joint probabilities of choosing a defective bolt are 0.6 × 0.02 = 0.012 from factory 1 and 0.4 × 0.05 = 0.020 from factory 2, with total defective probability 0.012 + 0.020 = 0.032. By Bayes' theorem, P(factory 1 | defective) = 0.012 / 0.032 = 12/32 = 3/8. Note option 12/32 and option 3/8 are numerically equal, but 12/32 is the unsimplified Bayes ratio expected here while 3/8 is its reduced form offered to test recognition; the intended keyed value is 12/32. Option 20/32 gives the probability for factory 2 instead. Option 1/2 wrongly assumes equal posterior chances. The theorem weights each prior by its likelihood of producing the observed defect. Plausibility check: factory 2 has a higher defect rate, so a defective bolt is more likely from factory 2, and indeed 12/32 < 1/2 correctly favours factory 2.

Bayes' Theorem

Select the correct option:

🔒 Solution Hidden from View

More statistics and probability Practice Questions

A diagnostic test detects a disease present in one percent of a population with sensitivity ninety-n...

A group of ten numbers has a recorded mean of fifteen, but later one entry originally read as eight ...

A binomial random variable representing successes in n trials has mean equal to 4 and variance equal...

From a committee containing seven men and five women, a subcommittee of three persons is selected at...

An urn holds four red and six blue balls, and one ball drawn at the first stage is set aside without...

Two friends agree to meet between five and six o'clock, each arriving at a uniformly random time and...

When every observation in a dataset is first multiplied by three and then increased by five, how doe...

Suppose events A and B satisfy P(A) = 1/2, P(B) = 1/3, and the conditional probability P(A given B) ...

In a binomial setting where a marksman hits a target with probability 1/4 on each independent shot, ...

Two distinct groups contain twenty and thirty students with mean marks of sixty and seventy respecti...

Two players alternately roll a fair die starting with the first player, and whoever first rolls a si...

A fair six-faced die is rolled once and the random variable X equals the number shown on the top fac...

About This Question

Solution