Tree Diagrams

Build the tree from the experiment

Consider two draws without replacement from a bag containing A1, A2, A3, B1 and B2. At each draw, every remaining token is equally likely. Record only whether each token has an A or B label.

Start with branches A and B, labelled $3/5$ and $2/5$. After A, the bag has two A and two B tokens left, so the next branches have probabilities $2/4$ and $2/4$. After B, it has three A and one B, giving probabilities $3/4$ and $1/4$.

Outgoing branches at each reached node must describe disjoint possibilities covering what can happen next. Their conditional probabilities therefore total one. They do not need to be the same at different nodes.

Distinguish a branch probability from a path probability

The branch A after B has probability $3/4$. It means the chance of A second given B first. The complete BA path has probability

The first factor accounts for the chance of reaching the B-first node. Reading $3/4$ as the overall probability of BA would forget that factor.

The four complete paths are

Their total is one because they form a partition of the two-draw experiment. Four leaves do not imply four equal probabilities.

Add disjoint paths to answer an event question

Exactly one A corresponds to AB or BA:

At least one A corresponds to AA, AB or BA, with total $9/10$. The complement BB has probability $1/10$, providing an independent way to check that result.

For an A on the second draw, add AA and BA, again obtaining $3/5$. This is a marginal probability: the first result is not specified. The tree makes it possible to keep a stage label fixed while summing over the histories that reach it.

Read conditional information backwards

Suppose one token of each type was drawn, but their order is unknown. Among the two compatible paths AB and BA, each has probability $3/10$. The probability that A came first is therefore

This calculation first selects the compatible leaves and then rescales their probabilities. It does not replace the initial branch label by a guess. In less symmetric examples, the compatible leaves can have very different probabilities, so counting leaves alone is unreliable.

Change the branch labels when the procedure changes

With replacement and fresh independent selection, the second-stage probabilities would be $3/5$ and $2/5$ after either first result. The resulting AA, AB, BA and BB path probabilities are $9/25$, $6/25$, $6/25$ and $4/25$.

The same tree shape can therefore represent different experiments. What changes is the model attached to the branches. A useful tree caption states the starting composition, replacement rule and random-selection assumption, not just the topic name.

Trees can end at different depths

Toss an independent fair coin until the first head, but stop after at most three tosses. The terminal paths are

Their probabilities are $1/2$, $1/4$, $1/8$ and $1/8$, which total one. The shorter paths have greater probabilities because fewer events must occur to complete them.

The probability of seeing a head within the three-toss limit is $1/2+1/4+1/8=7/8$. The probability that the experiment uses all three tosses is $P(TTH)+P(TTT)=1/4$. These are different events, even though both refer to the stopping rule.

Audit a probability tree

Check that each branch label is conditional on the history leading to it. Confirm that children of a node are mutually exclusive and exhaustive. Multiply along complete paths, then verify that all terminal path probabilities total one. Finally, inspect the requested event to make sure every relevant path has been included exactly once.

A tree may compress outcomes into categories, as AA compresses six labelled-token outcomes here. That compression is legitimate because the branch probabilities retain the underlying weights. Branch lengths and vertical spacing are not probability scales unless a diagram explicitly says otherwise.

For many repeated trials, a full tree becomes unwieldy. Combinations and Binomial Probability can count equivalent paths without drawing each one.

Related topics

Conditional Probability develops reverse-condition calculations further. Counting Principle uses branching structure to count outcomes before probabilities are assigned.