Probability trees

When a tree is the right tool

• A probability tree is the representation for situations that happen in stages: a first event, then a second, then possibly a third.
• Use a tree when the question gives you conditional probabilities — "if it rains, the probability the ferry is cancelled is 0.4" — because those are exactly what the second-stage branches hold.

The two rules

• Multiply along a path to get the probability of that complete sequence of events.

• Add between paths when more than one path gives the outcome you want.

• Formally, multiplying along a path is the conditional probability rule rearranged:

$$P(A \text{ and } B) = P(A) \times P(B \mid A)$$

• The second-stage branches are conditional probabilities. The number on a branch from node $A$ is $P(B \mid A)$ — the probability of $B$ given that $A$ has already happened.

Checks that catch errors

• The branches at every node sum to 1. If they do not, you have made an error or misread a probability.
• All the complete path probabilities sum to 1, because the paths are mutually exclusive and exhaustive.
• Do both checks. They take seconds and catch almost everything.

With and without replacement

• This distinction changes the second-stage probabilities and is very frequently examined.

• Without replacement, reduce both parts of the fraction. Drawing from 12 items of which 5 are red: after drawing a red, the next draw has 4 red out of 11.
• Read the wording carefully. "Two students are chosen" almost always means without replacement — the same student cannot be chosen twice.

"At least one" on a tree

• Use the complement rather than adding several paths:

$$P(\text{at least one}) = 1 - P(\text{none})$$

• On a tree, "none" is usually a single path, so this converts three or four multiplications and an addition into one multiplication and a subtraction.

Trees with unequal stages

• The stages need not have the same number of branches, and a branch may end early if the process stops.
  • A quality inspection that stops as soon as a fault is found produces a tree where one branch terminates at each stage.
• Label every branch with its probability and every endpoint with the outcome. An unlabelled tree earns little, because the representation is itself part of what is assessed.

Trees versus tables

• The two representations hold the same information in different forms, and questions sometimes require converting between them.
  • Tree → table: each complete path probability becomes a cell of the table.
  • Table → tree: first-stage branches are the marginal probabilities; second-stage branches are the conditionals, found by dividing each cell by its row total.
• If a question gives conditional probabilities and asks for a reversed conditional, the usual route is: build the tree, compute the path probabilities, then divide. That is the subject of the next page.