How do I combine two independent events?

For independent events, P(A and B) is P(A) times P(B). P(A or B) is P(A) plus P(B) minus P(A and B), which avoids double-counting the overlap.

What does the Bayes mode calculate?

It computes the posterior P(A given B) from the likelihood P(B given A), the prior P(A) and the false-positive rate P(B given not A), using Bayes theorem. This is the classic test-accuracy calculation.

What is the complement of a probability?

The complement is the chance the event does not happen: P(not A) = 1 − P(A). If an event has probability 0.3, its complement is 0.7.

How do I enter a percentage as a decimal?

Divide the percentage by 100. So 25% is 0.25, 5% is 0.05 and 100% is 1. All inputs should be decimals between 0 and 1.

What does it mean for events to be independent?

Two events are independent if one happening does not change the probability of the other — like separate coin flips. The AND/OR mode assumes independence when multiplying the probabilities.

Why can a positive test still likely be wrong?

For a rare condition, even an accurate test produces many false positives relative to true positives. Bayes mode shows this: with a low prior, P(A given a positive) can be surprisingly low despite high test accuracy.

Are my inputs stored?

No. All probabilities are processed in your browser and nothing is transmitted.

Probability Calculator

Probability calculator

Three common calculations in one tool: the complement of a single event, the combined probability of two independent events with AND and OR, and Bayes’ theorem for a conditional probability. It covers the everyday probability questions in statistics, risk, testing and games of chance.

How it works

Pick a mode and enter probabilities as decimals between 0 and 1:

Complement:   P(not A) = 1 − P(A)
Independent:  P(A and B) = P(A) × P(B)
              P(A or B)  = P(A) + P(B) − P(A) × P(B)
Bayes:        P(A|B) = [P(B|A)·P(A)] /
                       [P(B|A)·P(A) + P(B|¬A)·(1 − P(A))]

The OR formula subtracts the overlap so the shared outcome is not counted twice. Bayes combines a prior, a likelihood and a false-positive rate.

Example

Two independent events with P(A) = 0.5 and P(B) = 0.4:

P(A and B) = 0.5 × 0.4 = 0.20
P(A or B) = 0.5 + 0.4 − 0.20 = 0.70

Bayes — a disease with prior P(A) = 0.01, a test with P(B|A) = 0.99 and false positive P(B|¬A) = 0.05:

P(A|B) = (0.99 × 0.01) / (0.99 × 0.01 + 0.05 × 0.99) ≈ 0.167, so a positive test means only about a 17% chance of having the disease.

Mode	Inputs	Result
Complement	P(A)=0.3	0.7
AND	0.5, 0.4	0.20
OR	0.5, 0.4	0.70

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