Probability Tree Calculator

Instructions: Enter the probabilities as decimals (e.g., 0.5 for 50%). Stage 1 represents the first event, and Stage 2 represents the outcomes dependent on Stage 1.

Probability of Event A occurring [P(A)]:

P(B) if A occurred [P(B|A)]:

P(B) if A did NOT occur [P(B|A')]:

Calculation Results

Path Description	Probability (Decimal)	Probability (%)

Combined Probability of Event B occurring:

This represents the total likelihood of Outcome B regardless of what happened in Stage A.

Please ensure all probabilities are between 0 and 1.

Understanding Probability Trees

A probability tree is a visual representation of all possible outcomes of a series of events. It is particularly useful in conditional probability, where the outcome of the second event depends on what happened in the first event. Our calculator solves two-stage probability trees, which are common in statistics, medicine, and risk assessment.

Key Components of a Probability Tree

Stage 1 (Event A): The initial branching point where you define the probability of the first event happening (P(A)) and not happening (P(A')).
Branches: The lines connecting events. The sum of probabilities branching from a single point must always equal 1.0 (100%).
Conditional Probabilities: These are the "second stage" branches. For example, P(B|A) is the probability of B happening given that A has already occurred.
Joint Probabilities: These are the final results at the end of each path, calculated by multiplying the probabilities along that specific path.

The Mathematical Formulas

The calculator uses the following logic to determine the four possible outcomes of a two-stage tree:

        1. Both A and B occur: P(A ∩ B) = P(A) × P(B|A)

        2. A occurs, B does NOT: P(A ∩ B') = P(A) × (1 – P(B|A))

        3. A does NOT occur, B does: P(A' ∩ B) = (1 – P(A)) × P(B|A')

        4. Neither occurs: P(A' ∩ B') = (1 – P(A)) × (1 – P(B|A'))

Practical Example: Diagnostic Testing

Imagine a medical test for a rare disease. This is a classic application of a probability tree:

Stage 1: Does the patient have the disease? (P(A) = 0.01 or 1%).
Stage 2 (Positive Test): If they have the disease, the test is 99% accurate (P(B|A) = 0.99). If they do NOT have the disease, the test has a 5% false-positive rate (P(B|A') = 0.05).

By entering these into the calculator, you can find the probability that a person actually has the disease if they test positive (Bayes' Theorem foundation).

Why the Total Probability Matters

The "Total Probability of Event B" represents the "Law of Total Probability." It sums every path where Outcome B occurs. This is vital in business forecasting—for example, calculating the total chance of a project succeeding (Event B) based on two different economic scenarios (Stage A).