Bayes' theorem in real life

Bayes' theorem is used for real decisions: in medicine, in quality checks in factories, and in insurance. The method is always the same:

• List the causes (machines, groups of people, routes) and their chances P(Eᵢ). They must add to 1.
• For each cause, write the chance of the observed result, P(A|Eᵢ).
• Multiply each pair: these are the branches.
• Add all the branches to get P(A).
• Divide the branch you want by the total.

Turn percentages into decimals first: 5% = 0.05, 0.5% = 0.005.

A surprising example: a disease affects 1% of people. A test finds it in 90% of ill people, but also wrongly shows positive for 5% of healthy people. A person tests positive. How likely are they to be ill?

• Ill and positive: 0.01 × 0.9 = 0.009.
• Healthy and positive: 0.99 × 0.05 = 0.0495.
• P(ill | positive) = 0.009 ÷ (0.009 + 0.0495) = 0.009/0.0585 ≈ 0.15.

Only about 15%, even though the test sounds good. The healthy group is so much bigger that its few false alarms outnumber the true positives. This is why doctors repeat tests before deciding.

Present the answer in a small table (cause, P(Eᵢ), P(A|Eᵢ), product) in exams. The examiner can then follow every number, and you can spot a slip quickly.