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.

What is the Probability Calculator?

Free probability calculator. Find the complement of an event, combine two independent events with AND and OR, or apply Bayes theorem for a conditional probability. Runs entirely in your browser. It runs free in your browser on Gera Tools, with nothing uploaded.

Probability Calculator

Name: Probability Calculator
Creator: Gera Tools
License: https://creativecommons.org/licenses/by/4.0/

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.

Worked examples

Complement. If the probability of rain is P(A) = 0.3, the probability it does not rain is P(not A) = 1 − 0.3 = 0.7. The complement of any event always adds to 1.

Two independent events. You flip a fair coin (P(heads) = 0.5) and roll a die hoping for a six (P(six) = 0.167). The chance of both: 0.5 × 0.167 = 0.083, about 1 in 12. The chance of at least one: 0.5 + 0.167 − 0.083 = 0.583.

Bayes’ theorem — why positive tests can still be mostly wrong. Suppose a rare condition affects 1% of the population (prior P(A) = 0.01). A test correctly identifies affected people with 99% sensitivity (P(B|A) = 0.99) but has a 5% false-positive rate among healthy people (P(B|¬A) = 0.05).

Given a positive test, the probability of actually having the condition:

P(A|B) = (0.99 × 0.01) / (0.99 × 0.01 + 0.05 × 0.99)
       = 0.0099 / (0.0099 + 0.0495)
       ≈ 0.167

Only about 17% — despite the test being highly accurate. This is the base-rate fallacy in action: when the prior probability is low, even a good test produces more false positives than true positives in absolute terms.

Mode	Inputs	Result
Complement	P(A) = 0.3	0.70
AND	P(A)=0.5, P(B)=0.4	0.20
OR	P(A)=0.5, P(B)=0.4	0.70
Bayes	Prior=0.01, Sens=0.99, FP=0.05	≈ 0.167

When to use each mode

Complement is the fastest calculation in probability — if you know the chance of something happening, the chance of it not happening is always 1 minus that.
AND / OR for independent events applies when neither event influences the other: rolling dice, drawing cards with replacement, independent component failures in a reliability analysis.
Bayes mode is the right tool whenever you need to reason from a test result back to a hypothesis: medical screening, spam filtering, fraud detection, A/B test interpretation, or any situation where you start with a prior belief and update it given evidence.

Common mistakes

Using the AND formula for non-independent events. If events are correlated — weather on Tuesday and Wednesday, or two parts from the same production batch — the simple product formula is wrong. The AND mode is only valid when outcomes genuinely do not influence each other.
Forgetting the base rate in Bayes. Plug in the prior P(A) that reflects how common the condition actually is in the population being tested, not just how good the test is. The Bayes result is often counterintuitive precisely because people ignore the base rate.

Everything runs locally in your browser — no inputs are transmitted anywhere.