Bayes' TheoremJEE Main

Concept

Bayes' theorem reverses conditionals: from P(+|D) (test accuracy) to P(D|+) (what a positive actually means). The denominator is the total probability — true positives PLUS false positives. With a rare disease, false positives from the huge healthy pool can swamp true positives: a 95%-accurate test on a 2% prevalence gives only ~28% posterior. Intuition fails; the formula doesn't.

Key formula

Derivation

P(D|+) = P(D ∩ +)/P(+) by definition. Numerator: P(D)P(+|D). Denominator by the law of total probability over the partition {D, ¬D}.

Count version (n = 10 000): nD = n·P(D) sick, of whom sens·nD test positive; (n − nD)·fpr healthy also test positive. Posterior = true positives / all positives.

Scenarios to explore

• Bayes' Theorem — Reverse the conditional — why rare-disease positives mislead.

Real-world applications

• Medical screening interpretation (why doctors retest).
• Spam filters: P(spam | words) from P(words | spam).
• Factory QC: which machine produced the defective item?

JEE exam tips

• JEE pattern: urns/machines B₁…Bₙ with priors and conditional draws — posterior = branch/total.
• Draw the tree: multiply along branches, add the branches that match the evidence.
• Answer sanity check: posterior must lie between prior and 1 when the test supports D.

Common mistakes

• Confusing P(+|D) with P(D|+) — the prosecutor's fallacy.
• Ignoring the base rate (prior) entirely.
• Denominator missing the false-positive branch.

Exam traps to avoid

• 'Test is 95% accurate' may mean sensitivity ≠ specificity — read which is given.
• Sequential updates: yesterday's posterior is today's prior — don't reuse the original.