How do I know whether to multiply or add on a probability tree diagram?

Multiply when you move along a single path (one outcome AND another). Add when you combine separate paths that each satisfy the same condition (one outcome OR another). Think "along a path = multiply" and "across paths = add."

Probability Tree Diagrams — Definition, Formula & Examples

A probability tree diagram is a branching visual that shows every possible outcome of a sequence of events, with probabilities written on each branch. You multiply along the branches to find the probability of any specific chain of outcomes.

A probability tree diagram is a directed graph in which each node represents the state after an event, each edge (branch) is labeled with the probability of transitioning to the next state, and the probability of any complete path from root to leaf equals the product of the branch probabilities along that path. The sum of all leaf probabilities equals 1.

Key Formula

P(\text{path}) = P(\text{branch}_1) \times P(\text{branch}_2) \times \cdots \times P(\text{branch}_n)

Where:

$P(\text{path})$ = Probability of one specific sequence of outcomes (one root-to-leaf path)
$P(\text{branch}_i)$ = Probability written on the i-th branch along that path
$n$ = Number of events in the sequence

How It Works

Start at a single point on the left and draw one branch for each possible outcome of the first event, labeling each branch with its probability. From the end of every branch, draw new branches for the second event's outcomes and label those probabilities too. To find the probability of a specific sequence, multiply the probabilities along its path from start to finish. If a question asks for the probability of multiple sequences (for example, "at least one heads"), add the probabilities of all paths that satisfy the condition.

Worked Example

Problem: A bag contains 3 red and 2 blue marbles. You draw one marble, do NOT replace it, then draw a second marble. What is the probability of drawing red first and blue second?

Step 1 — Draw the first set of branches: The first event has two outcomes: Red (3 out of 5) or Blue (2 out of 5).

P(R_1) = \frac{3}{5}, \quad P(B_1) = \frac{2}{5}

Step 2 — Draw the second set of branches (no replacement): After drawing red first, 4 marbles remain (2 red, 2 blue). Branch from R₁ into R₂ and B₂.

P(R_2 \mid R_1) = \frac{2}{4}, \quad P(B_2 \mid R_1) = \frac{2}{4}

Step 3 — Multiply along the desired path: The path "Red then Blue" runs through the R₁ branch and then the B₂ branch.

P(R_1 \text{ then } B_2) = \frac{3}{5} \times \frac{2}{4} = \frac{6}{20} = \frac{3}{10}

Answer: The probability of drawing red first and blue second is

\frac{3}{10}

, or 0.3.

Another Example

Problem: You flip a fair coin twice. Use a tree diagram to find the probability of getting at least one heads.

Step 1 — Build the tree: First flip: H or T, each with probability 1/2. Second flip: again H or T from each branch, each 1/2. This gives four paths: HH, HT, TH, TT.

Step 2 — Find each path probability: Multiply along each path. Every path has probability 1/2 × 1/2 = 1/4.

P(\text{each path}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

Step 3 — Add paths that satisfy "at least one H": The paths HH, HT, and TH all contain at least one heads. Add their probabilities.

P(\text{at least one H}) = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4}

Answer: The probability of getting at least one heads in two flips is

\frac{3}{4}

Why It Matters

Probability tree diagrams appear heavily in middle-school and high-school statistics courses, especially when events are dependent and replacement rules change the numbers. They also show up in real decision-making: doctors use similar branching diagrams to weigh the likelihood of test results, and game designers use them to balance the odds of in-game events.

Common Mistakes

Mistake: Using the same probabilities on the second set of branches even when the problem says "without replacement."

Correction: After removing an item, the total count drops by one and the category count may change. Recalculate each branch probability with the updated numbers.

Mistake: Adding probabilities along a single path instead of multiplying.

Correction: Moving along a path means both events must happen, so you multiply. You only add when combining the results of different paths.

Related Terms

Tree Diagram — General branching diagram; probability trees add branch probabilities
Dependent Events — Events where branch probabilities change after earlier outcomes
Fundamental Counting Principle — Counts total outcomes that tree branches display
Experimental Probability — Found by trials; tree diagrams show theoretical probability
Combination — Counts selections when order doesn't matter
Permutation — Counts ordered arrangements shown by tree paths
Combinatorics — Broader field of counting that includes tree methods