How to Draw a Probability Tree Diagram

Updated 10 November 2026 · Standard formula, shown below

This guide shows you how to draw a probability tree diagram step by step. You’ll learn how to structure branches for sequential events, write probabilities on each branch, and calculate the probability of combined outcomes by multiplying along the paths. No prior knowledge beyond basic fractions and percentages is needed.

What You’ll Need

Paper and a pencil (or a whiteboard/digital drawing tool)
A ruler (optional, for straight lines)
A clear understanding of the problem: what events happen, in what order, and what the possible outcomes are for each
Basic arithmetic skills (multiplying fractions or decimals)

Step-by-Step

Identify the sequential events. Read the problem and list the events in the order they occur. For example, “tossing a coin twice” has two events: first toss, then second toss. “Drawing two marbles without replacement” has two events where the second event’s probabilities depend on the first.
List all possible outcomes for the first event. Write these outcomes as labels on separate branches. For a coin, the outcomes are Heads and Tails. For a die, they are 1 through 6.
Draw the first set of branches. Start from a single point (the “root”) on the left. Draw one line (branch) for each outcome of the first event. Spread them evenly so there is room for further branches.
Write the probability on each first branch. Place the probability (as a fraction, decimal, or percentage) directly on the branch, near the label. For a fair coin, each branch gets 1/2. For a fair six-sided die, each gets 1/6. Ensure the probabilities of all branches from the same node sum to 1.
For each outcome of the first event, list the possible outcomes of the second event. These become the second set of branches, emanating from the end of each first branch. If the second event’s probabilities are independent (like a second coin toss), they are the same for every branch. If they are conditional (like drawing without replacement), they differ depending on which first branch you are on.
Draw the second-level branches. From the tip of each first branch, draw one new branch for each possible outcome of the second event. Label each with its outcome and its probability. Make sure that for each node, the probabilities of its outgoing branches sum to 1.
Repeat for any further events. Continue adding levels of branches until you have represented every event in the sequence. The final tips of the branches are the endpoints.
Calculate the probability of each path. Multiply the probabilities along the branches that lead to a particular endpoint. For example, if the first branch has probability 1/2 and the second has 1/3, the path probability is (1/2) × (1/3) = 1/6. Write this product at the end of the path.
Check that all path probabilities sum to 1. Add up the probabilities at all endpoints. If they do not sum to 1, you have a mistake—either a branch probability is wrong or you missed a branch.
Use the tree to answer the question. To find the probability of a specific combined outcome, read the product at its endpoint. To find the probability of a compound event (e.g., “at least one head”), add the probabilities of all paths that satisfy the condition.

Common Mistakes

Branch probabilities not summing to 1. Every node’s outgoing branches must sum to 1. If they don’t, you’ve missed an outcome or mislabeled a probability.
Forgetting to adjust probabilities for “without replacement”. When an item is not replaced, the denominator decreases by 1 for the next draw. For example, drawing two red marbles from a bag of 3 red and 2 blue: first red is 3/5, but second red (given first was red) is 2/4, not 2/5.
Multiplying incorrectly. Always multiply along the branches, not add. Only add when you are combining separate paths for a compound event.
Drawing branches too close together. This leads to labels overlapping and confusion. Give each branch enough space.
Using percentages and fractions inconsistently. Stick to one format throughout the tree to avoid arithmetic errors.

Tips

Use a ruler for straight lines—it makes the tree easier to read and less likely to have accidental mislabeling.
Write probabilities as simplified fractions (e.g., 1/2 not 2/4) to keep the multiplication simple.
Color-code branches by event or by condition (e.g., all “success” branches in one color) to quickly identify relevant paths.
For problems with more than two events, draw the tree in stages and check each node’s probabilities before moving on.
Always verify that the sum of endpoint probabilities equals 1—this catches most errors.

Troubleshooting

If your tree isn’t working, try these checks:

Your endpoint probabilities don’t sum to 1. Revisit each node and confirm that the probabilities on its branches add to 1. Then re-multiply each path.
You’re unsure about conditional probabilities. Write out the sample space after the first event. For “without replacement”, the second event’s denominator is one less. For “with replacement”, it stays the same.
The tree looks too crowded. Redraw with more space between branches. You can also split the tree into two separate trees if the events are independent (but a single tree is usually clearer).
You get a probability greater than 1 or negative. That indicates an arithmetic error or a branch probability that’s too large. Probabilities must be between 0 and 1.
You’re not sure if the events are independent. Check if the outcome of the first event affects the probabilities of the second. If it does, the tree must reflect that with different branch probabilities.

Frequently Asked Questions

Do I have to draw a tree for every probability problem?

No. Trees are most useful for problems with two or three sequential events, especially when conditional probabilities are involved. For a single event, a simple list or Venn diagram may suffice.

Can I use decimals instead of fractions?

Yes. Decimals work fine, but fractions often make the multiplication and simplification easier. Just be consistent throughout the tree.

What does “without replacement” mean for the tree?

It means the item is not put back, so the total number of items decreases. This changes the probabilities on the second-level branches. For example, drawing two cards from a deck without replacement: the first card’s probability is out of 52, the second out of 51.

How do I find the probability of “at least one” using a tree?

Identify all paths that contain at least one occurrence of the desired outcome, then add their endpoint probabilities. Alternatively, subtract the probability of the complementary event (none) from 1.

Why do the probabilities on branches from the same node always sum to 1?

Because the outcomes are mutually exclusive and exhaustive—exactly one of them must happen. The sum of their probabilities equals the probability of the certain event, which is 1.

Can a probability tree handle more than two events?

Yes, you just add more levels. The tree grows exponentially, so for many events it becomes large. For three or four events it’s still manageable; beyond that, consider other methods like formulas or simulation.

Method: the standard formula for this calculation, rounded as each result states. See how the calculators work or report an error.