Tree Diagram Probability: Step-by-Step Guide

Updated 8 November 2026 · Standard formula, shown below

This guide explains how to build and use a tree diagram to solve probability problems. A tree diagram helps you visualize all possible outcomes of a sequence of events and calculate the probability of combined outcomes. You’ll learn how to draw the branches, assign probabilities, and multiply along paths to find final probabilities.

What You’ll Need

Paper and pencil (or a digital drawing tool)
A basic understanding of fractions and decimals
Knowledge of how to multiply fractions (for calculating branch probabilities)
A calculator (optional, for checking arithmetic)

Step-by-Step

Identify the events and their order. Determine the sequence of events you’re analyzing. For example, if you’re flipping a coin twice, the first event is the first flip, and the second event is the second flip. Write down each event in order.
List all possible outcomes for the first event. For each outcome, draw a branch from the starting point. Label each branch with the outcome name (e.g., Heads, Tails).
Write the probability of each first-event outcome on its branch. The probability is a number between 0 and 1, often written as a fraction, decimal, or percentage. For a fair coin, each branch gets 1/2.
For the second event, draw branches from the end of each first-event branch. Each first-event outcome leads to a new set of branches representing the possible outcomes of the second event. Label these branches with the outcome names.
Write the conditional probability on each second-event branch. This is the probability of that outcome given that the first event’s outcome (the branch you’re coming from) has occurred. For independent events (like coin flips), the probabilities are the same for each branch. For dependent events (like drawing cards without replacement), the probabilities change based on the first outcome.
Continue for additional events. If there are more than two events, repeat steps 4 and 5 for each subsequent event, branching from every endpoint of the previous event.
Calculate the probability of each path (final outcome). Multiply the probabilities along the branches of each path from the start to the endpoint. For example, if you flip a coin twice, the probability of Heads then Tails is (1/2) × (1/2) = 1/4.
Check that all path probabilities sum to 1. Add up all the final probabilities. If they don’t sum to 1, you’ve missed a branch or made an arithmetic error.
Use the path probabilities to answer your question. If you need the probability of a specific combination, just read the product you calculated for that path. If you need the probability of multiple outcomes (e.g., at least one Heads), add the probabilities of all paths that meet that condition.

Common Mistakes

Forgetting to update probabilities for dependent events. When drawing without replacement, the probability on the second branch changes. For example, drawing a red marble from a bag of 5 red and 5 blue marbles: first draw red = 5/10, but the second draw red (without replacement) = 4/9. Using 5/10 again is wrong.
Not labeling branches clearly. If you don’t write the outcome names, you’ll confuse paths later.
Multiplying incorrectly. Always multiply the probabilities along the entire path, not just the last branch.
Adding when you should multiply. Multiply along a path (for AND), add across paths (for OR). Mixing these up is a classic error.
Ignoring that probabilities must be between 0 and 1. If you get a probability like 1.5, you’ve made a mistake.

Tips

Use fractions and simplify at the end. Keeping fractions like 1/2 × 1/2 = 1/4 is easier than using decimals (0.5 × 0.5 = 0.25) and avoids rounding errors.
Draw neatly and space branches evenly. This makes it easier to trace paths.
Write probabilities as decimals or percentages if that’s how the problem states them. Just be consistent.
For complex problems, break the tree into smaller parts. For example, if you have three events, first draw the tree for the first two events, then extend it.
Double-check that each set of branches from a single node sums to 1. For a fair coin, 1/2 + 1/2 = 1. For a die, 1/6 + 1/6 + … = 1.

Troubleshooting

If your tree diagram isn’t working, try these fixes:

Your path probabilities don’t sum to 1. Check that you’ve included every possible outcome at each branch. Missing a branch is the most common cause.
You’re getting probabilities greater than 1. You likely multiplied incorrectly or used a probability greater than 1 on a branch. Review each branch’s probability.
You’re unsure whether events are independent or dependent. Ask: Does the outcome of the first event affect the probability of the second? If yes, they’re dependent, and you must adjust the second branch probabilities accordingly.
You can’t find the probability of a compound event like “at least one.” Instead of adding many paths, consider the complement: 1 minus the probability of the opposite event (e.g., no heads). This often simplifies the calculation.
Your tree is too large. If you have many events, the tree grows exponentially. For more than 3 or 4 events, consider using other methods (like binomial probability) but the tree still works conceptually.

Frequently Asked Questions

When should I use a tree diagram instead of a formula?

Use a tree diagram when you have two or more sequential events and you need to visualize all possible outcomes. It’s especially helpful for dependent events where conditional probabilities are involved. For simple independent events with many repetitions, a binomial formula might be faster.

How do I handle probabilities that change after each event?

For dependent events, update the probability on each branch based on the previous outcome. For example, drawing two cards without replacement: the first card’s probability is out of 52, the second out of 51, and the specific counts change based on what was drawn first.

Can I use percentages on the branches?

Yes, but convert them to decimals (or fractions) when multiplying. For example, 50% × 50% = 0.25 = 25%. Just ensure all branches from a node sum to 100%.

What does the sum of all path probabilities equal?

The sum of the probabilities of all final outcomes (paths) must equal 1 (or 100%). This is a good check for your diagram.

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

Find the probability of the complement (none of the event happening) by following the path where the event doesn’t occur, then subtract from 1. For example, for at least one head in two flips, calculate the probability of Tails, Tails (1/4) and subtract from 1 to get 3/4.

What if my tree diagram has too many branches?

If you have many events or many outcomes per event, the diagram becomes unwieldy. Try combining outcomes or using a probability formula instead. For example, for rolling a die 4 times, a tree would have 6^4 = 1296 paths—impractical. Use binomial or multinomial formulas instead.

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