Probability Tree Calculator

Use this Probability Tree Calculator to enter values, adjust options, and review results in a compact responsive workspace.

Results are calculated automatically as you enter data.

Parent: Start

Branch outcome Probability

Tree limit: at most 100 total nodes and at most 12 levels, including the Start root.

Current branches

Parent	Outcome	P

Tree

Click a node to add children or inspect its path.

Selected path probability

Simulation mode

Trials Seed

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Assumptions and limitations

Results are estimates for educational and exploratory use only and are not professional advice. No guarantee is made that outputs are accurate, complete, current, suitable, or error-free. Verify inputs, units, formulas, assumptions, rounding, data quality, model fit, and applicable rules independently. Do not rely on this calculator as the only basis for financial, legal, medical, safety, academic, business, official, eligibility, or regulated decisions; consult a qualified professional when stakes are material. Use of this calculator is subject to our Terms of Use.

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Related Calculators

What Is a Probability Tree?

A probability tree is a branching diagram for a situation that unfolds in steps. Each branch represents a possible outcome at that step, and each branch is labeled with a probability. By following a path from the start of the tree to an ending point, you can model a complete sequence of outcomes.

Probability trees are especially useful when later outcomes depend on earlier outcomes. For example, the chance of traffic may be different on rainy days than on clear days. A tree lets you show that relationship clearly: first branch by weather, then branch again by traffic condition.

The main idea is simple:

Probabilities on the branches from the same parent should describe all possible next outcomes for that point in the tree.
A complete path probability is found by multiplying the probabilities along that path.
If the leaf paths cover all possible final outcomes, their probabilities should add to $1$, or $100\%$.

Why Probability Trees Matter

Probability problems can become confusing when there are several stages, conditional probabilities, or many possible final outcomes. A tree diagram helps by keeping the structure visible.

Probability trees are useful for:

understanding conditional probability in statistics classes;
comparing possible paths in games, experiments, surveys, and decision problems;
checking whether a probability model is complete;
showing the difference between a single branch probability and the probability of a full sequence;
connecting theoretical probabilities with simulated results.

A tree also helps prevent a common mistake: treating a later probability as if it were unconditional. In a tree, a second-stage branch is usually the probability of an outcome given the path that came before it.

Key Terms to Know

Outcome: A possible result, such as “Rain,” “No rain,” “Traffic,” or “Clear roads.”
Branch: One possible next outcome from a parent node.
Parent node: The point where one or more branches begin.
Leaf node: A terminal outcome with no further branches. A complete path ends at a leaf.
Branch probability: The probability assigned to one branch. In this calculator, it is entered as a decimal from $0$ to $1$.
Conditional probability: The probability of an event when another event or previous outcome is already known.
Joint probability: The probability that all outcomes on a path occur together.
Theoretical probability: The probability calculated from the branch values in the model.
Empirical probability: The observed frequency from repeated simulation trials.

How Probability Trees Work

A probability tree starts at a root, then branches into possible outcomes. Under each parent node, the probabilities of the child branches should add to $1$ when those branches cover all possible next outcomes.

For a parent with child branch probabilities $p_1, p_2, \ldots, p_m$, a complete branch group should satisfy:

$$ p_1 + p_2 + \cdots + p_m = 1 $$

To find the probability of a complete path, multiply the probabilities on the branches in that path:

$$ P(\text{path}) = \prod_{i=1}^{k} p_i $$

Where:

$p_i$ is the decimal probability of the $i$-th branch on the path.
$k$ is the number of branches in the path.
$P(\text{path})$ is the joint probability of that full sequence.

For a two-stage event, this is the familiar multiplication rule:

$$ P(A \text{ and } B) = P(A) \times P(B \mid A) $$

This means the probability of both $A$ and $B$ is the probability of $A$ multiplied by the probability of $B$ after $A$ has happened.

If a tree has several terminal paths, the total probability of all leaf outcomes is:

$$ T_{\text{leaf}} = \sum_{\ell \in L} P(\ell) $$

Where $L$ is the set of leaf paths. In a complete tree, this total should be close to $1$, or $100\%$.

Simulation adds another layer. If a path occurs $n_\ell$ times in $N$ simulation trials, its empirical probability is:

$$ E_\ell = \frac{n_\ell}{N} $$

The theoretical probability comes from the tree. The empirical probability comes from random trials, so it can vary from run to run.

Examples of Probability Trees in Practice

Example 1: A Simple Two-Branch Start

Suppose a test result can be either Pass or Fail:

Pass: $0.6$
Fail: $0.4$

The branch group is consistent because:

$$ 0.6 + 0.4 = 1 $$

That means the two branches cover all possible outcomes at the start. The probabilities may also be read as $60\%$ and $40\%$.

Example 2: Weather and Traffic

Suppose the first step is weather:

Rain: $0.3$
No rain: $0.7$

Then traffic depends on the weather:

If Rain: Traffic $0.8$, Clear roads $0.2$
If No rain: Traffic $0.25$, Clear roads $0.75$

The probability of the path Rain → Traffic is:

$$ P(\text{Rain and Traffic}) = 0.3 \times 0.8 = 0.24 = 24\% $$

The probability of the path No rain → Traffic is:

$$ P(\text{No rain and Traffic}) = 0.7 \times 0.25 = 0.175 = 17.5\% $$

To find the total probability of traffic, add the terminal paths that end in traffic:

$$ P(\text{Traffic}) = 0.24 + 0.175 = 0.415 = 41.5\% $$

This shows why trees are helpful: the same final outcome can be reached through more than one path.

Example 3: An Incomplete Branch Group

Suppose a weather branch group is entered as:

Rain: $0.3$
No rain: $0.6$

The sum is:

$$ 0.3 + 0.6 = 0.9 $$

That leaves $0.1$, or $10\%$, unaccounted for. The tree may still show the branches, but the model is incomplete until the missing probability is fixed or another branch is added.

Example 4: A Zero-Probability Branch

A branch probability of $0$ is allowed in the model. If a path includes a branch with probability $0$, the full path probability is also $0$:

$$ 0.5 \times 0 = 0 $$

A branch probability of $1$ means that, according to the model, that branch is certain from that parent. If one branch has probability $1$, the other sibling branches should sum to $0$ for the group to remain consistent.

How to Interpret the Result

The selected path result is the joint probability of following one exact sequence of branches from the start. It is not just the probability of the last outcome label; it includes every branch on the path.

The leaf outcome count tells you how many terminal paths are currently in the tree. This is a count, not a probability.

The leaf probability total shows the sum of all terminal path probabilities. In a complete and consistent tree, this should be close to $100\%$. If it is much lower or higher, the tree may have missing branches, duplicate coverage, or branch groups that do not add to $1$.

The consistency status checks whether each set of sibling branches adds to $100\%$. A consistent tree is important because each parent node should describe a complete set of possible next outcomes.

In simulation results, the theoretical column is based on the branch probabilities. The empirical column is based on random trials. With more trials, empirical results often get closer to theoretical probabilities, but they do not have to match exactly in any single simulation.

Common Mistakes and Misconceptions

Entering percentages instead of decimals. A probability of $60\%$ should be entered as $0.6$, not $60$ and not 60%.

Using fractions when decimals are required. A value such as $\frac{3}{5}$ should be entered as $0.6$.

Forgetting that sibling branches should add to $1$. If branches under the same parent add to $0.85$ or $1.20$, the tree does not represent a complete probability split.

Confusing branch probability with path probability. A branch might have probability $0.8$, but the full path to that branch could have probability $0.3 \times 0.8 = 0.24$.

Adding when you should multiply. Multiply along one path. Add separate terminal paths only when they represent alternative ways for the event of interest to occur.

Simulating before the tree is consistent. Simulation only makes sense when each branch group is a valid probability split.

Rounding too early. Rounding intermediate probabilities can slightly change final results. Keep decimal inputs as accurate as possible, then round the final interpretation.

Assuming a short graph label changes the real label. Long labels may be shortened in the graph view, but the full label is still the meaningful outcome name in tables and paths.

When to Use Probability Trees

Use a probability tree when a situation has stages or conditional outcomes, such as:

drawing items with or without replacement;
modeling weather followed by travel conditions;
describing pass/fail outcomes followed by next actions;
comparing possible routes through a decision or experiment;
teaching dependent and independent events;
checking whether a probability model covers all final outcomes;
comparing theoretical probabilities with simulated frequencies.

For a single one-step event, a tree may be unnecessary. For multi-step events, especially when probabilities change after earlier outcomes, a tree can make the structure much easier to understand.

Limitations and Things to Keep in Mind

A probability tree is only as reliable as the probabilities entered into it. If the branch probabilities are guesses, outdated, or based on poor data, the result will reflect those weaknesses.

This calculator uses decimal probabilities from $0$ to $1$. It does not parse entries such as 60% or 3/5. Convert those values to decimals before entering them.

Branch groups are checked for consistency, but they are not automatically normalized. If a group adds to $0.9$, the calculator reports the issue rather than changing the values for you.

Displayed probabilities are rounded as percentages, so a displayed result may be approximate. Very small rounding differences can occur, especially in trees with many decimal probabilities.

Simulation results are random. Two simulations with the same tree and trial count can produce different empirical percentages. A larger number of trials usually reduces random variation, but it does not remove it completely.

The trial count for simulation is limited to whole numbers from $1$ through $100000$. Very small trial counts can be useful for demonstration, but they may differ noticeably from the theoretical probabilities.

To keep the diagram usable, a tree can contain at most 100 total nodes and at most 12 levels, including the Start root. Add branches or import rows in parent-before-child order within those limits.

For important decisions involving money, health, safety, legal obligations, engineering work, or official records, treat probability tree results as an educational or modeling aid. Double-check the assumptions and consult a qualified professional when needed.

How to Use This Calculator

Select the parent node where you want to add a branch. The tree starts with Start selected.
Enter a branch outcome label, such as “Rain” or “Pass.”
Enter the branch probability as a decimal from $0$ to $1$.
Add the branch under the selected parent.
Repeat until all sibling branches under each parent add to $1$.
Review the selected path result, leaf outcome count, leaf probability total, and consistency status.
Edit branch labels or probabilities in the branch table when needed.
Remove a branch if you also want to remove its descendant subtree.
Use import only when your rows are in parent, outcome, probability order and each parent appears before its children. Legacy rows require unique outcome labels; Start always refers to the root, even if a child outcome is also named Start. Use four-column IDs when labels repeat.
Enter an integer number of trials from $1$ to $100000$ before running a simulation.
Compare theoretical and empirical percentages after simulation.
Download the graph as a PNG if you need an image of the current tree.

Frequently Asked Questions

Why do the branches under a parent need to add to 1?

Branches under the same parent represent the possible next outcomes from that point. If they cover all possibilities, their probabilities should add to $1$, or $100\%$. If they do not, the tree is either missing probability, double-counting probability, or using probabilities that do not belong in the same branch group.

Do I multiply or add probabilities in a probability tree?

Multiply probabilities along a single path to find the probability of that exact sequence. Add probabilities across separate terminal paths when those paths are alternative ways for the event you care about to happen.

What is the difference between a branch probability and a joint probability?

A branch probability describes one step from one parent node. A joint probability describes a complete path, so it combines every branch probability along that path. For example, $0.3$ may be the probability of rain, while $0.3 \times 0.8 = 0.24$ is the joint probability of rain and traffic.

Why does the simulation not exactly match the theoretical probability?

Simulation uses random trials. The empirical percentage is based on what happened in those trials, so it can be above or below the theoretical probability. With more trials, the empirical value often gets closer to the theoretical value, but exact agreement is not guaranteed.

Can I enter 60% or a fraction such as 3/5?

Use decimals only. Enter $60\%$ as $0.6$, and enter $\frac{3}{5}$ as $0.6$. Percentage signs and fraction strings are not treated as valid probability inputs.

Can a branch probability be 0 or 1?

Yes. A probability of $0$ means the branch contributes no probability to any path that uses it. A probability of $1$ means the branch is certain from that parent in the model, so the other sibling branches should sum to $0$.

Sources and References

Books

Barbara Illowsky, Susan Dean, et al. Introductory Statistics 2e. OpenStax, 2023. Chapter 3, “Probability,” especially Section 3.5, “Tree and Venn Diagrams,” and Chapter 3 Key Terms. https://openstax.org/books/introductory-statistics-2e/pages/3-5-tree-and-venn-diagrams; https://openstax.org/books/introductory-statistics-2e/pages/3-key-terms
Charles M. Grinstead and J. Laurie Snell. Introduction to Probability, 2nd ed. American Mathematical Society, 2003; freely redistributable version by Peter G. Doyle, 2006. Chapters 4 and 8. https://math.dartmouth.edu/~prob/prob/prob.pdf

Online and Official Sources

LibreTexts Statistics. “The Monte Carlo Simulation V2.” Accessed July 4, 2026. https://stats.libretexts.org/Bookshelves/ComputingandModeling/RTG%3ASimulatingHighDimensionalData/TheMonteCarloSimulationV2