Probability Calculator
Work out the probability of two or three events happening together or separately, update a probability with Bayes’ theorem, or convert between probability, odds and “1 in N” — with a Venn diagram and every step of the working.
Written by Suhaib Hassan. Every rule used here is a standard theorem, and each worked example is re-derived by hand. Our methodology.
It checks your inputs are valid probabilities, then applies the multiplication rule, the addition rule, the complement rule and Bayes’ theorem exactly as written below.
Mutually exclusive and independent are often confused. This page makes the relationship an explicit choice and rejects combinations that are impossible.
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Kolmogorov’s three axioms
All of modern probability rests on three rules, formalised by Andrey Kolmogorov in 1933. This calculator rejects any input that breaks them.
1. Non-negativity
The probability of any event is never negative.
2. Unit measure
Something in the sample space must happen, so the total probability is exactly 1.
3. Additivity
For events that cannot occur together, probabilities add.
Mutually exclusive vs independent events
Two of the most confused ideas in probability. They are close to opposites.
| Property | Mutually exclusive | Independent |
|---|---|---|
| Meaning | A and B cannot both happen: A ∩ B = ∅ | Knowing A happened tells you nothing about B |
| Intersection P(A ∩ B) | 0 | P(A) × P(B) |
| Union P(A ∪ B) | P(A) + P(B) | P(A) + P(B) − P(A)P(B) |
| Conditional P(A | B) | 0 | P(A) |
| Can both apply? | Only if P(A) = 0 or P(B) = 0. If both events are possible, being mutually exclusive means knowing one happened tells you the other did not — so they are dependent. | |
Worked examples
Each is checked by hand and can be reproduced above.
Rare-condition screening
A condition affects 1 in 1,000 people. A test detects it 95% of the time and gives a false positive 5% of the time. Someone tests positive — what is the chance they have it?
P(A|B) = 0.00095 ÷ 0.05090
= 0.0187 (about 1.87%)
Most positives are false positives because the condition is so rare. Use the Bayes’ theorem tab.
At least one six in four rolls
Rolling a fair die four times: the chance of at least one six.
P(no six in four) = (5/6)⁴ = 0.4823
P(at least one six) = 1 − 0.4823 = 0.5177
Use the Odds tab with p = 1/6 (0.1667) and 4 trials.
Two independent backups
Each of two independent backup systems fails with probability 0.02.
At least one fails: 0.02 + 0.02 − 0.0004 = 0.0396
Enter 0.02 and 0.02 on the Two events tab, independent.
What is probability?
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It can be written as a decimal, a percentage, a fraction, odds, or a frequency such as “1 in 20”.
The core rules
- Complement: P(not A) = 1 − P(A).
- Addition (either event): P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The intersection is subtracted so it is not counted twice.
- Multiplication (both events), independent: P(A ∩ B) = P(A) × P(B).
- Conditional probability: P(A | B) = P(A ∩ B) ÷ P(B), for P(B) > 0.
- Bayes’ theorem: P(A | B) = P(B | A) × P(A) ÷ P(B), where P(B) = P(B | A)P(A) + P(B | not A)P(not A).
- Odds: odds in favour = p ÷ (1 − p); p = a ÷ (a + b) for odds a : b.
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Sources
- Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin.
- Grinstead, C. M. & Snell, J. L. (1997). Introduction to Probability, 2nd ed. American Mathematical Society.
- Gigerenzer, G. & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: frequency formats. Psychological Review 102(4), 684–704.
Frequently asked questions
What is the difference between mutually exclusive and independent events?
Mutually exclusive events cannot happen together, so P(A ∩ B) = 0. Independent events can happen together, and one occurring does not change the probability of the other, so P(A ∩ B) = P(A) × P(B). If both have non-zero probability, events cannot be both.
Why can’t I just add probabilities for “A or B”?
If the events can overlap, adding counts the overlap twice. The addition rule subtracts P(A ∩ B) once to correct this. Adding alone is only right for mutually exclusive events.
What does Bayes’ theorem do?
It updates a prior probability when you observe new evidence. It reverses a conditional probability: from P(B | A), how likely the evidence is if A is true, to P(A | B), how likely A is given the evidence, taking into account how common A was to begin with.
What is the gambler’s fallacy?
It is the mistaken belief that past independent outcomes change future ones — for example that heads is “due” after a run of tails. For independent events, P(A | B) = P(A): the coin has no memory.
How do I convert probability to odds?
Odds in favour are p ÷ (1 − p). A probability of 0.4 gives odds of 0.4 ÷ 0.6 = 2 to 3 in favour, or 3 to 2 against. Use the Odds tab to convert in either direction.
Can a probability be greater than 1 or negative?
No. Every probability lies between 0 and 1 inclusive (0% to 100%). The calculator rejects values outside that range, and joint probabilities that are impossible given P(A) and P(B).
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