Bayes' theorem

Bayes' theorem turns a conditional probability around:

P(A | B) = P(B | A) × P(A) ÷ P(B)

The medical test puzzle

A disease affects 1% of people. A test finds 99% of sick people (sensitivity), and wrongly says "positive" for 5% of healthy people. You test positive. How likely are you to be sick?

Most people guess about 99%. Count 10,000 people instead:

• 100 are sick; 99 of them test positive.
• 9,900 are healthy; 5% of them, 495, also test positive.
• Positives: 99 + 495 = 594. Sick among them: 99.

P(sick | positive) = 99 ÷ 594 ≈ 0.167: only about 1 in 6.

people = 10000
prevalence, sensitivity, false_alarm = 0.01, 0.99, 0.05

sick = people * prevalence
true_pos = sick * sensitivity
false_pos = (people - sick) * false_alarm
print("true positives:", round(true_pos), " false positives:", round(false_pos))
print("P(sick | positive) =", round(true_pos / (true_pos + false_pos), 3))

The base rate (how rare the disease is) matters enormously. Because healthy people are so many, their few false alarms outnumber the true positives. Doctors confirm a positive screening with a second, different test for this reason.

Common mistakes

• Ignoring the base rate (the base rate fallacy).
• Treating P(positive | sick) and P(sick | positive) as the same number.

Mark lesson complete