Tree diagram

The tree diagram is the standard graphical tool for visualising and computing probabilities in multi-stage experiments. Fundamental convention: conditional probabilities always appear on the branches of the tree, never loose numbers. More precisely:

• at the first level, the branches leave the root and carry the events of the first experiment, with the weights $P(A_1), P(A_2), \ldots, P(A_n)$ of the marginal (absolute) probabilities;
• at the second level, from each node $A_i$ branches depart carrying the events $B_j$ with weight $P(B_j \mid A_i)$, the conditional probability of $B_j$ having already observed $A_i$;
• the probability of a leaf is the product along the path: $P(A_i \cap B_j) = P(A_i) \cdot P(B_j \mid A_i)$ (this is precisely the product formula, now read geometrically on the tree).

Two-level tree for a diagnostic test: at the first level the disease $M$/$\overline{M}$ with marginal probabilities, at the second level the test outcome $T^+$/$T^-$ with conditional probabilities; in red the leaf probabilities.

Reading. At the first level the population is split into $M$/$\overline{M}$ with marginal probabilities $P(M) = 0,001$ and $P(\overline{M}) = 0,999$. At the second level, for each branch, it splits into $T^+$/$T^-$ with the conditional probabilities: $P(T^+ \mid M) = 0,99$, $P(T^+ \mid \overline{M}) = 0,02$, and so on. The probability of a “leaf” event is the product of the probabilities along the path: $P(M \cap T^+) = 0,001 \cdot 0,99 = 0,00099$ and $P(\overline{M} \cap T^+) = 0,999 \cdot 0,02 = 0,01998$ (in red beside the leaves).

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Topics: Probability
Concepts: Tree diagram · Product formula · Conditional probability
Methods: Tree diagram
Skills: Calculating probability · Modelling