Contingency Tables and Tree Diagrams

A contingency table is simply a grid that counts people by two traits at once, like whether they exercise and whether they sleep 7 hours. The name comes from "contingent," meaning "depending on": the table lets you see whether one trait depends on the other. The formula $P(A \mid B) = \frac{\text{count of } A \text{ and } B}{\text{count of } B}$ is less scary than it looks: the top is the people with both traits (one cell), and the bottom is everyone with trait $B$ (one row or column total). It's the conditional probability from the last lesson, done by counting instead of by formula.

Reading a table (200 people surveyed):

• Joint: $P(\text{exercises and sleeps 7+}) = \frac{60}{200}$.
• Marginal: $P(\text{exercises}) = \frac{90}{200}$, from a total.
• Conditional: $P(\text{sleeps 7+} \mid \text{exercises}) = \frac{60}{90}$: only the "exercises" row counts.

Notice that the conditional one matches the formula from the last lesson: $\frac{P(\text{both})}{P(\text{exercises})} = \frac{60/200}{90/200} = \frac{60}{90}$. The 200s cancel, which is why with a table you can skip straight to counts.

Tree diagrams: each branch carries a conditional probability. Multiply along a path for "and"; add the paths that lead to the event you want. Multiplying along a path is just the multiplication rule, one step at a time, and adding paths works because different paths can't both happen.

A common mistake. It's tempting to divide by the grand total, 200, for every question. That's only right for joint and marginal probabilities. A conditional probability like "given they exercise" means you only look at the 90 people who exercise, so 90 goes on the bottom.

Medical researchers summarize trials in two-way tables (treatment vs outcome); marketers cross-tab customer segments against purchases.