Study Bayes' theorem with a connected example

Start with the question, not the equation

Bayes' theorem relates a conditional probability to the reverse conditional probability and the relevant base rates. Penn State's probability lesson provides the formal treatment. For your study map, put a question at the center: “Given a warning, how likely is a printer jam?”

The following numbers are invented for practice. They are not measurements of a real printer or a prediction system.

A complete worked example

Suppose 10% of print jobs jam. A warning appears for 80% of jobs that jam, and for 10% of jobs that do not jam. Let J mean a jam and W mean a warning.

• Prior: P(J) = 0.10; P(not J) = 0.90.
• Likelihood: P(W | J) = 0.80.
• False warning: P(W | not J) = 0.10.
• Total warning probability: P(W) = 0.80 × 0.10 + 0.10 × 0.90 = 0.17.
• Posterior: P(J | W) = 0.08 / 0.17 ≈ 0.471, or 47.1%.

Across 1,000 hypothetical jobs, 100 would jam and 80 of those would have warnings. Of the other 900, 90 would have warnings. That gives 80 jammed jobs among 170 warned jobs. This count-based check reaches the same result.

Make each branch explain one role

Create branches for “Before the warning,” “Warning behavior,” “All warnings,” and “After the warning.” Put the calculation inside the corresponding node's expanded notes. This keeps a page of symbols from overwhelming the map.

Add a “Common mistakes” branch: reversing the conditional, forgetting false warnings, or using 0.80 as the final answer. Write why each error changes the question rather than only writing that it is wrong.

For mathematical notation in Mappy, use the formula control. You can retain the plain-language explanation next to the rendered equation, and keep a short code example in another note if you want to check the arithmetic separately.

Change one assumption

Hold the warning behavior fixed but change the prior jam rate to 1%. The numerator becomes 0.008 and the denominator becomes 0.107, giving about 7.5%. Explain why the answer changes before checking your calculation.

That final exercise makes the map more than a formula sheet. It asks you to understand which assumptions drive the result. Use your course's notation consistently, and keep rounding until the end of the calculation.