# Conditional Probability Calculator

URL: https://speedycalculator.com/calculators/statistics/conditional-probability/
Last updated: 2026-10-30

> The conditional probability calculator builds a full probability tree from P(A), P(B | A) and P(B | not A) – or from P(A), P(B) and P(A and B) – and returns P(A | B), P(A | not B), every joint probability and the total probability of B.

## Inputs

- What do you know?
- P(A) (%)
- P(B | A) (%)
- P(B | not A) (%)
- P(A) (%)
- P(B) (%)
- P(A and B) (%)

## Results

- P(A | B)
- Probability tree
- P(A | not B)
- P(not A | B)
- P(not A | not B)
- P(B)
- P(not B)
- P(A and B)
- P(A and not B)
- P(not A and B)
- P(not A and not B)
- P(not A)
- P(B | A)
- P(not B | A)
- P(B | not A)
- P(not B | not A)
- Are A and B independent?

## Formula

```
P(A | B) = P(A and B) ÷ P(B)
P(B) = P(A) × P(B | A) + P(not A) × P(B | not A)
P(A and B) = P(A) × P(B | A)
```

## Explanation

A conditional probability is the chance of one event once you know another has happened. P(A | B) restricts attention to the outcomes where B occurred and asks what share of them also have A. The defining formula is P(A | B) = P(A and B) ÷ P(B).

In practice you rarely know P(A and B) and P(B) directly. You usually know how likely A is, and how likely B is in each branch: a test result given disease and given no disease, a defect rate for each of two machines, or rain chances for each type of morning. The calculator draws this as a two-level tree, multiplies along each branch to get the four joint probabilities and adds the branches that end in B to get P(B). Reversing the conditioning then gives P(A | B), P(A | not B) and their complements.

If you already have P(A), P(B) and the overlap P(A and B), choose the second mode: the calculator fills in the same tree from those numbers and also tells you whether the events are independent.

### How to use the Conditional Probability Calculator

- **What do you know?**
- **P(A)**
- **P(B | A)**
- **P(B | not A)**
- **P(A)**
- **P(B)**
- **P(A and B)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Each leaf of the tree is the product of the probabilities on its path. P(B) is the sum of the leaves that end in B. Dividing a leaf by P(B) or P(not B) turns it into a reverse conditional probability (Bayes’ rule).

### Worked example

5% of a population is infected. A test is positive for 91% of infected people and for 5% of healthy people (95% specificity). Joint probabilities: infected and positive = 0.05 × 0.91 = 0.0455; healthy and positive = 0.95 × 0.05 = 0.0475. P(positive) = 0.0930. P(infected | positive) = 0.0455 ÷ 0.0930 = 48.92%. A negative result leaves P(infected | negative) = 0.0045 ÷ 0.907 = 0.50%.

### Reading the tree

The first fork splits on A, the second on B. The four numbers on the right are the joint probabilities and always add to 100%. To reverse the conditioning, pick the leaves that end in the event you observed and see what share of them came through the A branch.

## Worked example

Inputs — What do you know?: P(A), P(B | A) and P(B | not A); P(A): 5 %; P(B | A): 91 %; P(B | not A): 5 %.

Results — **P(A | B) = 48.9247 %**; P(A | not B) = 0.4961 %; P(not A | B) = 51.0753 %; P(not A | not B) = 99.5039 %; P(B) = 9.3 %; P(not B) = 90.7 %.

## Key terms

- **Conditional probability**: The probability of an event given that another event has occurred, P(A | B).
- **Joint probability**: The probability that two events both occur, P(A and B).
- **Marginal probability**: The probability of one event on its own, such as P(B), found by adding branches.
- **Independence**: Knowing one event does not change the probability of the other.

## FAQ

**How do you calculate conditional probability?**
Divide the probability of both events by the probability of the condition: P(A | B) = P(A and B) ÷ P(B). With P(A and B) = 18% and P(B) = 30%, P(A | B) = 60%.

**Is P(A**
B) the same as P(B | A)? | No. In the medical example P(positive | infected) = 91% but P(infected | positive) is only 48.92%.

**How do I know if two events are independent?**
They are independent when P(B | A) = P(B | not A) = P(B), or equivalently P(A and B) = P(A) × P(B).

**What is a probability tree?**
A diagram whose first branches split on A and whose second branches split on B. Multiplying along a path gives a joint probability, and all leaves add to 1.

**What is the law of total probability?**
P(B) = P(A)P(B | A) + P(not A)P(B | not A) – the chance of B averaged over both branches.

**Can a conditional probability be larger than P(A)?**
Yes. If B is evidence for A, P(A | B) exceeds P(A); in the example a positive test raises 5% to 48.92%.


## Sources

- [OpenStax – Introductory Statistics 2e, Chapter 3: Probability topics](https://openstax.org/books/introductory-statistics-2e/pages/3-introduction)
- [Penn State STAT 414 – Introduction to Probability Theory](https://online.stat.psu.edu/stat414/)
- [Khan Academy – Statistics and probability](https://www.khanacademy.org/math/statistics-probability)