Single-event probability, joint AND, union OR, conditional and complement — all on one page, with a Venn diagram and step-by-step working.
Enter P(A) and P(B) as decimals (0–1) or percentages.
P(A or B) — at least one occurs
P(A and B)
0.20
joint / intersection
P(A | B)
0.50
A given B
P(B | A)
0.40
B given A
P(neither)
0.30
1 − P(A∪B)
P(only A)
0.30
A and not B
P(only B)
0.20
B and not A
P(exactly one)
0.50
symmetric difference
Odds for A
1 : 1
favourable : against
| Formula | Meaning | When |
|---|---|---|
| P(A∩B) = P(A)·P(B) | Joint probability, multiplication rule | independent |
| P(A∩B) = 0 | Cannot both happen | mutually exclusive |
| P(A∪B) = P(A) + P(B) − P(A∩B) | At least one — inclusion-exclusion | always |
| P(A | B) = P(A∩B) / P(B) | Conditional probability | always (P(B)>0) |
| P(Aᶜ) = 1 − P(A) | Complement | always |
| P(A | B) = P(A) | B tells you nothing about A | independent |
The Rules
Three simple rules cover most everyday probability work. The complement rule says P(not A) = 1 − P(A). The addition rule (inclusion-exclusion) says P(A or B) = P(A) + P(B) − P(A and B) — the last term avoids double-counting the overlap. The multiplication rule for two independent events is P(A and B) = P(A) · P(B); for dependent events it generalises to P(A and B) = P(A) · P(B | A), where P(B | A) is the conditional probability of B given A.
Working for your events
About This Tool
A probability calculator takes the chances of two events — A and B — and works out every combined probability that follows: their joint intersection (A and B), their union (A or B), each conditional probability (A given B, B given A), the complements (not A, not B), the chance that exactly one occurs, that neither occurs, and the equivalent odds.
Two events can relate to each other in different ways, and the rules change accordingly. Independent events (coin flips, separate dice rolls) satisfy P(A∩B) = P(A) · P(B). Mutually exclusive events (rolling a 3 vs. a 5 on a single die) cannot both occur, so P(A∩B) = 0. In real-world data, events often have a custom overlap somewhere between these extremes — which is why this tool lets you specify P(A∩B) directly.
From these basic inputs, the calculator applies Kolmogorov's axioms and the standard inclusion-exclusion identity to derive every related probability automatically. The Venn diagram visualises the relationships and updates live with your inputs.
Use this free probability calculator for stats coursework, risk analysis, A/B testing intuition, card-game probabilities, insurance and reliability calculations, or anywhere you need to combine probabilities cleanly. All computation runs locally — no sign-up, no tracking, no data leaves your device.
AND / OR / Conditional
Every combined probability from P(A) and P(B) in one view.
Custom Overlap
Specify P(A∩B) directly when events are neither independent nor exclusive.
Live Venn Diagram
Circles update as you change inputs — see the overlap visually.
Step-by-Step Working
Inclusion-exclusion and conditional rules laid out explicitly.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Odds Conversion
Returns "for : against" odds alongside probability — useful for betting maths.
Two inputs, one relationship choice, every combined probability.
Type each event's probability as a decimal between 0 and 1. So 25% is 0.25, 70% is 0.7. The sliders make it easy to scan a range.
Independent (most common) means one doesn't affect the other. Mutually exclusive means they can't both occur. Custom lets you supply the overlap directly.
The headline number is the probability at least one of A or B occurs — exactly what most everyday questions ("either / or") are asking for.
Stat tiles show P(A and B), the conditionals P(A | B) and P(B | A), plus P(neither) and P(exactly one).
Two circles, sized to your inputs, show how much overlap exists between A and B. Bigger overlap = larger P(A∩B), and a larger conditional probability.
The formula card shows every substitution — perfect for showing your work in a homework or exam answer.
Everything you need to know about combining probabilities — and the common pitfalls.
Probability is a number between 0 and 1 that measures how likely an event is. 0 means impossible; 1 means certain; 0.5 means equally likely to occur or not. For equally-likely outcomes, P(event) = favourable outcomes / total outcomes — so rolling a 3 on a fair six-sided die has probability 1/6 ≈ 0.167.
P(A AND B) is the probability that both events occur. For independent events it's P(A) × P(B); in general it's P(A) × P(B | A). P(A OR B) is the probability that at least one occurs — by inclusion-exclusion, P(A) + P(B) − P(A AND B). The subtraction prevents counting the overlap twice.
Two events are independent if knowing whether one occurred tells you nothing about whether the other did. Tossing a coin twice gives independent flips. Drawing cards from a deck without replacement does not — the second draw's probabilities depend on what happened on the first. Formally: A and B are independent iff P(A∩B) = P(A) · P(B) (equivalently P(A | B) = P(A)).
P(A | B) — read "the probability of A given B" — is the probability of A among only the outcomes where B occurs. Formally: P(A | B) = P(A∩B) / P(B). If A and B are independent, P(A | B) = P(A); knowing B happened doesn't change the chance of A. Conditional probability is the engine of Bayesian reasoning.
The complement of A, written Aᶜ or A', is the event "A does not occur". P(Aᶜ) = 1 − P(A). The complement is often the easier way to compute "at least one" — for example, the probability of at least one head in 5 coin flips is 1 − (1/2)⁵ = 31/32.
Mutually exclusive events cannot both occur — they have empty intersection, so P(A∩B) = 0. Rolling a 3 and rolling a 5 on a single die are mutually exclusive. For such events, the addition rule simplifies to P(A∪B) = P(A) + P(B) (the overlap correction is zero).
Probability is favourable outcomes divided by total outcomes — always between 0 and 1. Odds are favourable to unfavourable — so probability 0.25 corresponds to odds of 1:3 (one favourable for every three unfavourable). Conversion: probability = a / (a+b) for odds a:b; odds = p / (1−p).
The first Kolmogorov axiom requires probabilities to be non-negative; the second requires the certain event (everything that can happen) to have probability 1. Combined, every individual event must have probability between 0 (impossible) and 1 (certain). Values outside this range are not probabilities.
No. The intersection is a subset of each event individually, so P(A∩B) ≤ min(P(A), P(B)). The calculator clamps your custom intersection to this maximum. The minimum is max(0, P(A) + P(B) − 1) — set by the constraint that P(A∪B) cannot exceed 1.
Bayes' theorem reverses a conditional probability: P(A | B) = P(B | A) · P(A) / P(B). It's the basis of Bayesian inference — updating beliefs about A given new evidence B. Famous example: if a test for a rare disease (prevalence 1%) is 99% accurate, the probability you actually have the disease given a positive test is only about 50% — Bayes saves you from overreacting to a positive result.
Internally everything works in decimals (0–1). If you have a percentage, simply divide by 100 — 30% → 0.30. The calculator will accept any number, but values must satisfy 0 ≤ P ≤ 1 to be meaningful probabilities.
A permutation/combination calculator counts how many ways an arrangement or selection can happen. A probability calculator takes those counts (or any probabilities) and combines them. Often you need both: count outcomes to get base probabilities, then combine them. Use our permutation calculator for the counting step.