Probability tree
Complete all parent groups to validate the full tree.
Dashed labels are complement-solved values.
Probability Tree Calculator
A free probability tree calculator for multi-stage and conditional probability. Enter branch probabilities as decimals, fractions, or percentages — the built-in probability tree solver fills in any missing branch, and every path probability updates in real time.
Path Results
Select paths to add event probabilities.
Selected probability: 0
How to Use the Probability Tree Calculator
Define outcomes, enter conditional probabilities, check the tree, then select paths to add event probabilities.
Define outcomes
Create 2-3 stages and name 2-4 outcomes in each stage.
Enter probabilities
Type decimals, fractions, or percentages under each parent node — 0.3, 1/3, and 30% all work.
Check the tree
Every parent group is validated to sum to 100%, and a single blank branch is solved automatically.
Add paths
Select any valid paths in the results table to sum them into an event probability.
The Basics
What Is a Probability Tree?
The simplest way to organize multi-stage probability — one event, one set of branches at a time.
A probability tree — also called a probability tree diagram — is a visual map of a multi-stage random process. Each node is an event, each branch is one possible outcome labeled with its probability, and each root-to-leaf path is a complete sequence of outcomes. Multiply along a path to get its probability; add paths together to combine scenarios.
Nodes and Branches
Every node is an event; the branches leaving it are its possible outcomes, labeled with conditional probabilities. The branches from the same parent always sum to 1 — the calculator checks this for you.
Reading a Path
Follow the branches from left to right to read a sequence of outcomes. Multiply along a path to get its joint probability; add several paths to answer questions like 'at least one'.
Independent vs. Dependent Events
Coin flips keep the same probabilities at every stage; drawing without replacement changes them. The calculator stores probabilities per parent node, so both cases work.
The Math
Probability Tree Rules and Formulas
The calculator does the arithmetic for you — here is exactly what it computes.
Multiply Along Branches
P(A ∩ B) = P(A) × P(B|A)
The probability of a complete path is the product of its branch probabilities. This multiplication rule is the core of every probability tree calculation.
Add Across Paths
P(A or B) = P(A) + P(B)
For mutually exclusive paths, add their probabilities to find the chance that any of them happens — like 'at least one head in two flips'. Select paths in the results table and they are summed for you.
Every Parent Sums to 1
P(B₁) + P(B₂) + … = 1
The branches leaving any node must total exactly 1. Leave one branch blank and the probability tree solver fills it in as 1 minus the others.
Classic Problems
Worked Examples
Three classic problem types, solved the way this probability tree calculator computes them.
Independent Events: Rolling Two Dice
Roll a fair die twice and track only whether each roll is a six. A six-sided die has too many faces to list individually, so group the outcomes: Six and No Six, with the same probabilities at both stages. Enter 1/6 for Six and leave No Six blank — the solver fills in 5/6 for you.
| Path | Calculation | Probability |
|---|---|---|
| Six → Six | 1/6 × 1/6 | 1/36 ≈ 0.0278 |
| Six → No Six | 1/6 × 5/6 | 5/36 ≈ 0.1389 |
| No Six → Six | 5/6 × 1/6 | 5/36 ≈ 0.1389 |
| No Six → No Six | 5/6 × 5/6 | 25/36 ≈ 0.6944 |
P(at least one six) = 1 − 25/36 = 11/36 ≈ 0.3056 — or select the three paths containing Six and the calculator sums them.
Dependent Events: Two Cards Without Replacement
Draw two cards from a standard 52-card deck without putting the first back. Stage 1 is Heart 1/4 and Not Heart 3/4. Stage 2 changes per parent — after a Heart, 12 of the 51 remaining cards are hearts; after a non-Heart, 13 of 51 are. Enter every value as a fraction; the calculator converts automatically.
| Path | Calculation | Probability |
|---|---|---|
| Heart → Heart | 1/4 × 12/51 | 1/17 ≈ 0.0588 |
| Heart → Not Heart | 1/4 × 39/51 | 13/68 ≈ 0.1912 |
| Not Heart → Heart | 3/4 × 13/51 | 13/68 ≈ 0.1912 |
| Not Heart → Not Heart | 3/4 × 38/51 | 19/34 ≈ 0.5588 |
P(at least one heart) = 1 − 19/34 = 15/34 ≈ 0.4412. Fractions like 12/51 are entered exactly as written — no decimal conversion needed.
Total Probability: Quality Control
A factory's Machine A produces 60% of items with a 2% defect rate; Machine B produces 40% with a 5% defect rate. Each parent node gets its own probabilities — that is what makes the tree a conditional probability calculator.
| Path | Calculation | Probability |
|---|---|---|
| Machine A → Defective | 0.6 × 0.02 | 0.012 |
| Machine A → Passed | 0.6 × 0.98 | 0.588 |
| Machine B → Defective | 0.4 × 0.05 | 0.020 |
| Machine B → Passed | 0.4 × 0.95 | 0.380 |
P(defective) = 0.012 + 0.020 = 0.032 — a 3.2% overall defect rate, found by selecting the two paths ending in Defective.
Missing Branches
Probability Tree Solver: Find a Missing Branch
Leave exactly one branch under a parent blank, and the solver fills it in as 1 minus the sum of its siblings — no algebra required.
One Blank Branch
The solver triggers only when exactly one branch under a parent is empty and every sibling is a valid number.
The Complement Rule
missing = 1 − Σ siblings
Because every parent's branches must sum to 1, the missing value is whatever makes the total 100% — even 0 is a valid solution.
Solved, Not Stored
A solved value recalculates automatically when you change a sibling — until you type your own value into the blank branch.
Example: One Branch Left Blank
A stage has three outcomes. You know P(Red) = 0.40 and P(Blue) = 0.30, but not P(Green) — so leave Green empty.
| Branch | You Enter | Result |
|---|---|---|
| Red | 0.40 | 0.40 |
| Blue | 0.30 | 0.30 |
| Green | (blank) | solved: 0.30 |
Solved values are visually marked in the tree and update whenever a sibling changes.
missing = 1 − (0.40 + 0.30) = 0.30. If the siblings already exceed 1, the group is flagged instead of producing an impossible answer.
Probability Tree Calculator FAQ
Common questions about conditional probability trees, solving missing branches, and path calculations.
- What probability formats can I enter?
What probability formats can I enter?
Decimals (0.3), fractions (1/3), and percentages (30%) — in any combination. Each entry is parsed, shown with its equivalent decimal, and used directly in the path calculations.
- Can this probability tree solver find a missing probability?
Can this probability tree solver find a missing probability?
Yes. Leave exactly one branch under a parent blank and it is filled in as 1 minus the sum of its siblings. Solved values update automatically when you change the other branches, and each parent group is solved independently.
- Does it work for dependent events like drawing without replacement?
Does it work for dependent events like drawing without replacement?
Yes. Probabilities are stored per parent node, not per stage, so 'After Red' and 'After Blue' can have completely different branch probabilities — exactly what without-replacement and conditional probability problems need.
- How do I calculate 'at least one' probabilities?
How do I calculate 'at least one' probabilities?
Build the tree, then tick every path that matches your event in the Path Results table — for 'at least one head', select all paths containing Heads. The calculator sums the selected paths into a single event probability.
- Does this calculator solve Bayes' theorem?
Does this calculator solve Bayes' theorem?
No. It calculates forward through the tree — multiplying along branches and adding selected paths — but does not reverse conditional probabilities. For a medical-test question you can still read P(sick | positive) off the path table: divide the 'Sick → Positive' path by the sum of all paths ending in Positive.
- How many stages and outcomes are supported?
How many stages and outcomes are supported?
Up to 3 stages with 2-4 outcomes each, for a maximum of 64 leaf paths. That covers coin flips, draws with and without replacement, medical tests, quality control, and most classroom problems.
Need a Freeform Probability Tree?
Try the AI probability tree generator when your scenario needs a custom diagram from a natural-language description instead of a structured calculator.