Probability Tree Generator
Create an interactive probability tree diagram for sequential events. Edit branches and probabilities, calculate every possible outcome, and explore conditional probability, independent events, and sampling with or without replacement.
Your probability tree
Edit the branches and generate a tree, or choose a preset.
Thick branches and diamond markers identify selected paths (the intersection for conditional events). Branch lengths do not represent probabilities. Scroll inside the diagram to inspect larger trees; full labels appear in the table.
Show all outcomes
| Path | Branch probabilities | Joint probability | Percentage |
|---|
Show conditional probabilities
Show calculation
How to use the Probability Tree Generator
- Choose a custom probability tree or a drawing-with-replacement example.
- Define the stages, outcome labels, and branch probabilities. Each set of branches leaving a node must total 100%.
- Generate the tree to see every complete outcome and its joint probability.
- Select paths or stage-level events to calculate combined and conditional probabilities.
- Open the probability table and calculation steps to understand how each result was obtained.
Worked example: drawing marbles without replacement
A bag contains three red marbles and two blue marbles. Two marbles are drawn one after another without replacement. What is the probability of drawing one red and one blue marble, in either order?
This example uses sample values, not your current inputs.
Step 1: First draw
P(Red first) = 3/5
P(Blue first) = 2/5
Step 2: Second draw
After a red marble is removed, two red and two blue marbles remain.
P(Blue second | Red first) = 2/4
After a blue marble is removed, three red and one blue marble remain.
P(Red second | Blue first) = 3/4
Step 3: Multiply along each qualifying path
P(Red then Blue) = (3/5) × (2/4) = 3/10
P(Blue then Red) = (2/5) × (3/4) = 3/10
Step 4: Add the qualifying paths
P(One red and one blue) = 3/10 + 3/10 = 3/5 = 0.6
60%
There is a 60% probability that the two draws contain one red marble and one blue marble, regardless of their order.
Probability tree formulas and rules
A probability tree diagram represents the possible outcomes of successive random events. Each branch has a probability, and each complete path represents one possible sequence. The total probability across all complete paths equals 1.
Multiplication rule
To calculate the probability of one complete path, multiply the probabilities on its branches. For two stages, P(A and B) = P(A) × P(B | A). The second probability is conditional on the first event.
Addition rule
When an event can occur through several mutually exclusive complete paths, add their path probabilities. For example, drawing one red and one blue marble in two draws includes the paths Red–Blue and Blue–Red.
Branch probabilities must sum to one
All possible outcomes leaving the same node must have probabilities that add up to 1. If the probabilities do not sum to 1, the tree is incomplete or contains inconsistent inputs.
Independent vs. dependent events
Events are independent when learning the outcome of one does not change the probability of the other. For repeated independent coin tosses, later branch probabilities remain the same. Dependent events require probabilities that reflect previous outcomes, such as drawing objects without replacement.
The shared-stage control explicitly repeats a distribution across histories. Equal probabilities entered separately are not automatically treated as an independence declaration.
Probability trees with and without replacement
With replacement, an object is returned before the next draw, so the original population is restored. Without replacement, the selected object is removed, changing the remaining quantities and often the probabilities of later outcomes.
Impossible depleted branches are omitted; a remaining single outcome is shown with probability 1.
Conditional probability and Bayes' theorem
Conditional probability measures the chance of an event given that another event occurred. Bayes' theorem uses prior and conditional probabilities to find a reverse conditional probability, such as the probability of a condition given a positive test result.
P(A | B) = P(A ∩ B) / P(B)
P(A | B) = P(B | A) × P(A) / P(B)
The denominator P(B) must be greater than zero. A zero-probability condition makes the conditional probability undefined, not zero.
Illustrative medical-test example
Suppose P(Condition) = 0.01, P(Positive | Condition) = 0.90 and P(Positive | No condition) = 0.05. The positive joint probabilities are 0.009 and 0.0495, so P(Positive) = 0.0585. Bayes' theorem gives P(Condition | Positive) = 0.009 / 0.0585 ≈ 0.1538461538, or 15.38%.
A positive result therefore does not imply a 90% posterior probability. This example illustrates base-rate effects using sample probabilities, not the reliability of any actual clinical test or clinical advice.
Related probability tools
Use the Venn Diagram Calculator for set unions and intersections, the Binomial Distribution Calculator for success counts in independent trials, or the Probability Distribution Calculator for a discrete outcome table.
Sources and review
- OpenStax — Tree and Venn Diagrams
- OpenStax — Conditional Probability and the Multiplication Rule
- OpenStax — Contingency Tables and Probability Trees
Reviewed: October 2026. Formulas, sample calculations and deterministic engine checked against independent rational fixtures. Read our methodology.
Common questions
What is a probability tree diagram?
A probability tree diagram shows the possible outcomes of a sequence of events. Each branch represents one outcome and has a probability. Multiplying probabilities along a complete path gives the probability of that sequence.
How do you make a probability tree diagram?
Start with the possible outcomes of the first event. Add branches for the next event after each previous outcome, and label each branch with the appropriate probability. Continue for the remaining stages. Every group of outgoing probabilities must add up to 1.
How do you calculate probability from a tree diagram?
Multiply the probabilities along a complete path to find its joint probability. If the event can happen through several mutually exclusive paths, add the probabilities of those paths.
Do the probabilities on a tree diagram have to add up to 1?
Yes. The probabilities leaving each node must add up to 1 because they represent all possible outcomes at that point. The probabilities of all complete paths must also total 1.
What is the difference between independent and dependent events?
Independent events do not change one another's probabilities. For dependent events, the probability of a later result depends on what happened earlier. A probability tree can represent either situation.
How does a probability tree work without replacement?
The selected object is removed after each draw. Later probabilities are calculated using the remaining objects. For example, after drawing one red marble from a bag of three red and two blue marbles, the next draw uses two red and two blue marbles.
What changes when sampling with replacement?
The object is returned after each draw, restoring the original category counts. When draws use the same unchanged population independently, their probabilities remain constant.
What is a joint probability on a probability tree?
A joint probability is the probability that a particular combination of events occurs. On a tree diagram, it is calculated by multiplying the conditional probabilities along the matching complete path.
How do you calculate conditional probability using a tree?
Find the total probability of paths satisfying both the target and conditioning events, then divide by the probability of the conditioning event. The formula is P(A | B) = P(A ∩ B) / P(B), provided P(B) is greater than zero.
Can I use a probability tree for Bayes' theorem?
Yes. A two-stage probability tree can display prior probabilities, conditional likelihoods, joint outcomes and posterior probabilities. Bayes' theorem is especially useful when finding the probability of an earlier event after observing a later outcome.
How many stages can a probability tree have?
A probability tree can represent multiple sequential events. This generator supports up to three stages, subject to its displayed complexity limit, and calculates the probabilities of complete paths.
What is the difference between a probability tree and a Venn diagram?
A probability tree organizes sequential events and their conditional outcomes. A Venn diagram shows relationships between sets, including unions and intersections. Both can help explain probability, but they represent different structures.
Can I calculate the probability of several outcomes together?
Yes. Select the complete paths that satisfy the event. The calculator adds their probabilities. Complete paths are mutually exclusive, so they can be combined without double-counting.
What if a conditional event has zero probability?
Conditional probability is undefined when the conditioning event has probability zero. The calculator should report that limitation instead of displaying an invented numerical result.
Use these tools for mathematical information, not as a substitute for checking assumptions. Read our methodology or disclaimer.