What is Bayes’ Theorem, and how would you explain it intuitively in an interview?

What you need to know

The formula

P(A | B) = P(B | A) × P(A) / P(B)P(B) = P(B | A) × P(A) + P(B | not A) × P(not A)

The second line expands the bottom of the fraction: evidence B can arise either because A is true or because it is not.

It comes straight from the multiplication rule. P(A and B) can be written two ways — P(A | B) × P(B) or P(B | A) × P(A). Set them equal and divide by P(B).

Worked example with counts, not formulas

A disease affects 1 in 1,000 people. The test catches 99% of real cases (sensitivity) and wrongly flags 1% of healthy people (false-positive rate). You test positive. What is the chance you have the disease?

Picture 100,000 people:

Have the disease:   100   → 99 test positive     (true positives)Healthy:         99,900   → 999 test positive    (false positives)All positives: 99 + 999 = 1,098P(disease | positive) = 99 / 1,098 ≈ 9%

Only 9%. The test is good, but there are so many healthy people that 1% of them (999) outnumbers all the sick people who tested positive (99). Counting people like this — natural frequencies — is the easiest way to explain Bayes in an interview.

1def posterior(prior, sensitivity, false_positive_rate):2    """P(disease | positive test) by Bayes' theorem."""3    evidence = sensitivity * prior + false_positive_rate * (1 - prior)4    return sensitivity * prior / evidence56print(round(posterior(0.001, 0.99, 0.01), 3))   # rare disease7print(round(posterior(0.10, 0.99, 0.01), 3))    # common disease, same test

0.090.917

The same test gives 9% or 92% depending only on the prior. That one comparison shows why base rates matter.

A real-life example

A bank's fraud model flags UPI payments. It catches 95% of fraud and flags 2% of genuine payments. Fraud is 1 in 2,000 payments. Out of 1,000,000 payments:

Fraud:       500   → 475 flaggedGenuine: 999,500   → 19,990 flaggedP(fraud | flagged) = 475 / 20,465 ≈ 2.3%

About 43 of every 44 flagged payments are genuine. If every flag blocks the payment, thousands of customers get angry every day. The team uses Bayes to set policy: flags trigger a cheap step-up check, such as an OTP, not a block, and they work on reducing the false-positive rate, which matters far more than raising recall.

An everyday version from cricket. You hear on the radio that a delivery "turned sharply" but missed who bowled it. The spinner bowls 20% of the overs and turns the ball sharply on 60% of deliveries. The seamers bowl 80% of the overs and get sharp turn on 5%. P(spinner | sharp turn) = 0.2 × 0.6 / (0.2 × 0.6 + 0.8 × 0.05) = 0.12 / 0.16 = 75%. Strong evidence, but a one-in-four chance it was a seamer remains, because seamers bowl so much more.

Follow-up questions to expect

• "How do you improve P(disease | positive)?" — Lower the false-positive rate, or test a group where the disease is more common (a higher prior), or run a second independent test on the positives.
• "What is P(B) called and why is it needed?" — The evidence or marginal likelihood; it rescales the result so the posteriors over all hypotheses add to 1.
• "How does this relate to precision?" — Precision is exactly P(actually positive | predicted positive), so it falls when the positive class is rare, even with a good model.