Tree Diagrams for Sequential Events

Why Use a Probability Tree?

When an experiment unfolds in stages, the probability of a later outcome may depend on what happened earlier. A probability tree makes that structure visible: each split represents possible outcomes at one stage, and each branch is labeled with the probability of taking it from the current point. Following branches from the start to a terminal point describes one complete sequence of outcomes.

This organization is useful for draws without replacement, multi-step processes, and repeated trials. It helps prevent a common error: using an unconditional probability for a branch when the relevant probability is conditional on the path already taken.

Definition: A probability tree is a diagram for a sequence of stages. Each node represents the history of outcomes so far; each outgoing branch represents a possible next outcome and is labeled by its probability conditional on that history. A complete path follows branches from the starting node to a terminal node and records one full sequence of outcomes.

How to Read and Label a Tree

Begin by specifying the stages and the possible outcomes at each stage. At the first split, branch labels are probabilities of the first-stage outcomes. At a later split, the labels describe the next outcome given the history that leads to that node. Thus, branches at different nodes can have different probabilities even when they describe the same kind of outcome.

At any node that has positive probability of being reached, the probabilities of its outgoing branches sum to 1, provided those branches list all possible next outcomes. This is because exactly one of those next outcomes must occur. A conditional probability at a node is defined only if the history represented by that node has positive probability. If a node has zero probability of being reached, there is no conditional branch probability to label there.

The probability of a complete path is the product of its branch probabilities. This multiplication follows from the multiplication rule for sequential events: each branch gives the probability of the next outcome given the preceding outcomes. A branch label by itself is not usually the probability of the complete path; it is conditional on reaching the node.

Worked Example: Two Draws Without Replacement

Suppose a bag contains 3 red and 2 blue tokens. Two tokens are drawn one after another without replacement. Construct the tree and find the probability that the first token is red and the second is blue.

At the first draw, the probabilities are \(P(R)=3/5\) and \(P(B)=2/5\). After a red token is drawn, 2 red and 2 blue tokens remain. After a blue token is drawn, 3 red and 1 blue token remain. The tree is:

First draw red, probability \(3/5\).
- Then red, probability \(2/4\) given first red: path \(RR\).
- Then blue, probability \(2/4\) given first red: path \(RB\).
First draw blue, probability \(2/5\).
- Then red, probability \(3/4\) given first blue: path \(BR\).
- Then blue, probability \(1/4\) given first blue: path \(BB\).

The requested event is the complete path \(RB\). Multiply the branch probabilities along that path:

P(RB)=P(R)P(B\mid R) =\frac{3}{5}\cdot\frac{2}{4} =\frac{6}{20} =\frac{3}{10}.

Thus, the probability of drawing red first and blue second is \(3/10\), or 0.300. The second-draw branch probability \(2/4\) is conditional on the first draw being red; it is not by itself the probability of the full sequence \(RB\).

The Multiplication Rule Along a Path

A tree’s path calculation is an application of the multiplication rule. For two successive events, if \(P(A)>0\), the definition of conditional probability gives:

P(B\mid A)=\frac{P(A\cap B)}{P(A)} \quad\Longrightarrow\quad P(A\cap B)=P(A)P(B\mid A).

The positive-probability condition matters: \(P(B\mid A)\) is defined by this formula only when \(P(A)>0\). In a tree, this means the node for the first event must be reachable with positive probability before a conditional branch from that node is labeled.

The same reasoning extends to longer paths. Let \(A_1,\ldots,A_k\) describe the successive outcomes along a path, and let \(H_j=A_1\cap\cdots\cap A_j\) be the history through stage \(j\). If each history \(H_1,\ldots,H_{k-1}\) has positive probability, then:

\begin{aligned} P(H_k) &=P(H_{k-1})P(A_k\mid H_{k-1})\\ &=P(H_{k-2})P(A_{k-1}\mid H_{k-2})P(A_k\mid H_{k-1})\\ &=\cdots\\ &=P(A_1)P(A_2\mid A_1)\cdots P(A_k\mid A_1\cap\cdots\cap A_{k-1}). \end{aligned}

Each equality applies the two-event multiplication rule to the history so far and the next outcome. The conditions ensure that every conditional probability used in the product is defined. If a branch has probability zero, its path has probability zero; no conditional probabilities need to be assigned to later branches beyond that impossible history.

Worked Example: A Three-Stage Path With Independent Trials

Imagine three independent flips of a hypothetical coin that has probability 0.6 of landing heads and 0.4 of landing tails on each flip. Find the probability of the sequence heads, tails, heads.

The first split has branches \(H\) with probability 0.6 and \(T\) with probability 0.4. At each later node, the same two branch probabilities apply because the flips are independent. Each history has positive probability, so the conditional branch probabilities are defined. In particular, the probability of tails on the second flip given a head on the first is 0.4, and the probability of heads on the third given the first two results is 0.6.

P(HTH) =P(H)P(T\mid H)P(H\mid H\cap T) =0.6\cdot0.4\cdot0.6 =0.144.

So the probability of this exact three-flip sequence is 0.144, or 14.4%. The tree represents eight terminal sequences in all, but this calculation concerns only the path \(HTH\).

Reading a Tree in Context

A branch probability answers a conditional question at a particular node. A complete path probability answers a joint question about the entire sequence up to a terminal point. Keeping those interpretations distinct makes it easier to translate a word problem into a tree and to explain the result in context.

For example, a statement such as “the probability of a defect given the evening shift” belongs on the branch for “defect” leaving the evening-shift node. The probability of “evening shift and defect” belongs to the complete path that starts with evening shift and then follows the defect branch. These are different quantities, even though the path uses the conditional probability as one of its factors.

Worked Example: Shift Followed by Defect Status

Suppose a hypothetical workshop has a randomly selected production item made during the day shift with probability 0.7 and during the evening shift with probability 0.3. The probability an item is defective is 0.02 given the day shift and 0.05 given the evening shift. Find the probability that a randomly selected item was made during the evening shift and is defective.

The first split is shift: day with probability 0.7 or evening with probability 0.3. At the evening node, the defect branch has probability 0.05 and the not-defective branch has probability \(1-0.05=0.95\). The evening node has positive probability, \(0.3\), so these conditional branch probabilities are defined.

P(\text{evening and defective}) =P(\text{evening})P(\text{defective}\mid\text{evening}) =0.30\cdot0.05 =0.015.

The probability is 0.015, or 1.5%. This is the probability of the complete evening-and-defective path among all selected items. The branch value 0.05 instead means that, among items made during the evening shift, the probability of a defect is 5%.

As a check on the meaning of the tree, the other branch from the evening node has probability 0.95. The two conditional outcomes at that node are defective and not defective, so their probabilities sum to 1.

A Reliable Construction Process

A careful tree separates the description of the experiment from the arithmetic. Use this sequence of decisions:

List the stages.
Write down what happens first, second, and later. Make sure each stage is expressed as an outcome or category relevant to the question.

Draw all possible branches.
At each node, include the possible next outcomes. Branches leaving the same node should describe mutually exclusive outcomes that cover the possibilities at that stage.

Label each branch conditionally.
Use the history represented by its node when identifying the probability. Check that the node has positive probability before using a conditional probability given that history.

Follow the requested path.
Translate the event in the question into the appropriate sequence of branches, then multiply the labels along that complete path.

State what the result means.
Identify the sequence whose probability you calculated and express the value in the units or context of the problem, such as a proportion or percentage.

Common Mistakes

Using the same probability at every node without justification. In sampling without replacement, the composition changes after a draw, so later branch probabilities can change. Use the history at the node to determine each conditional probability.
Confusing a branch with a path. A branch labeled \(P(B\mid A)\) gives the probability of \(B\) among cases where \(A\) has occurred. The probability of the path \(A\) then \(B\) is \(P(A)P(B\mid A)\).
Multiplying unconditional probabilities in a dependent process. The product \(P(A)P(B)\) gives the joint probability only under independence. In general, use \(P(A)P(B\mid A)\).
Labeling a branch after an impossible history. If the history has probability zero, the conditional probability given that history is not defined by the usual ratio. Do not assign a conditional branch probability there.
Leaving the event unclear. A numerical answer should be tied to the path or event it describes; “0.015” alone does not say whether it is a conditional probability or a complete-path probability.

Key takeaway: A tree organizes successive outcomes. A branch label is conditional on reaching its node, while the probability of a complete path is the product of the branch probabilities along that path, provided the required conditional probabilities are defined.

Check Your Understanding

For each question, identify the relevant branch or complete path before calculating.

A container has 4 green and 1 orange token. Two tokens are drawn without replacement. What outcomes should label the second-stage branches after a green token is drawn first?
In a tree, a first-stage event \(A\) has probability 0.4, and the next branch has probability 0.3 given \(A\). What is the probability of the complete path \(A\) followed by that branch?
Explain why the probability of a later outcome can differ between two nodes at the same stage.
Suppose \(P(A)=0\). Can the usual conditional probability \(P(B\mid A)\) be used to label a branch after \(A\)? Explain.
A branch from a node is labeled 0.08. What additional information is needed to interpret this number correctly: the probability of a complete path, or a conditional probability?