🎲 Probability Calculator
Enter the probability of two events and get every combination: both happening, either happening, neither, and exactly one. Works with independent events or with a known joint probability.
Combining probabilities, and the mistakes that follow
A probability is a number between 0 and 1 measuring how likely something is. The arithmetic for a single event is trivial. The arithmetic for two events depends entirely on how they relate to each other, and that relationship is where almost every error happens.
The three relationships
| Relationship | P(A and B) | P(A or B) |
|---|---|---|
| Independent | P(A) × P(B) | P(A) + P(B) − P(A)P(B) |
| Mutually exclusive | 0 | P(A) + P(B) |
| General case | P(A) × P(B given A) | P(A) + P(B) − P(A and B) |
The subtraction in the "or" column is the inclusion-exclusion principle: adding both probabilities counts the overlap twice, so you remove it once. Forgetting it is the most common error with "or" questions, and it can produce probabilities above 100%, which is a useful signal that something went wrong.
Independent is not the same as mutually exclusive
These get confused constantly and they are near opposites. Independent means one event tells you nothing about the other: two coin flips. Mutually exclusive means they cannot both happen: a single coin landing heads and tails. Mutually exclusive events are in fact strongly dependent, because knowing one happened tells you the other definitely did not.
The complement trick
For questions phrased as "at least one", calculating directly means summing many cases. Calculating the opposite is usually one line:
P(at least one) = 1 − P(none)
Roll a die four times: what is the chance of at least one six? Directly, that is four separate cases. By complement, the chance of no six in one roll is 5/6, so in four rolls it is (5/6)⁴ = 0.482, and the answer is 1 − 0.482 = 51.8%.
Conditional probability
P(A | B) = P(A and B) / P(B)
Read as "the probability of A given that B has happened". The denominator shrinks the universe to the cases where B occurred. This is the engine behind Bayes' theorem, and behind the most counterintuitive result in applied probability: a test that is 99% accurate for a disease affecting 1 in 10,000 people still produces a positive result that is wrong about 99% of the time, because the false positives from the healthy majority vastly outnumber the true positives.
The gambler's fallacy
After five reds on a roulette wheel, black is not "due". The wheel has no memory and each spin remains independent, so the probability is unchanged. The related error in the opposite direction, the hot hand fallacy, assumes a streak will continue. Both come from expecting short sequences to look like long-run averages, which they do not.
The birthday problem is the classic demonstration of how badly intuition handles this. In a room of 23 people, the chance that two share a birthday is just over 50%, because there are 253 possible pairs rather than 23 comparisons.
A worked example
Event A has probability 40%, event B 25%, and they are independent. Both happen: 0.4 × 0.25 = 10%. Either happens: 0.4 + 0.25 − 0.10 = 55%. Neither: 1 − 0.55 = 45%. Exactly one: 0.55 − 0.10 = 45%. And A given B: 0.10 / 0.25 = 40%, which is just P(A) again, exactly as independence requires.
Frequently Asked Questions (FAQs)
What is the difference between independent and mutually exclusive?
Independent means one event gives no information about the other, so both can happen together. Mutually exclusive means they cannot both happen. Mutually exclusive events are highly dependent, since one occurring guarantees the other did not.
How do I calculate the probability of at least one event?
Use the complement: 1 minus the probability that none of them happen. For independent events, that is 1 − (1−p)ⁿ where n is the number of trials. It is almost always faster than enumerating the cases.
What are odds versus probability?
Probability is favourable outcomes divided by all outcomes. Odds are favourable against unfavourable. A probability of 1/4 is odds of 1 to 3 against. Betting markets quote odds; statistics quotes probability, and converting between them trips people up regularly.
Why is the birthday problem so surprising?
Because people compare themselves against the other 22 people rather than counting all 253 possible pairs. The probability of a shared birthday in a group of 23 is 50.7%, and in a group of 50 it is 97%.
What is conditional probability used for?
Anything where new information changes the picture: medical testing, spam filtering, insurance pricing, fault diagnosis. It is the basis of Bayes' theorem, which formalises how a prior belief should be updated by evidence.
Does a run of bad luck make good luck more likely?
No, for independent events. That is the gambler's fallacy. A roulette wheel, a die and a coin have no memory. It can be true for dependent events, such as drawing from a deck without replacement, where removing cards genuinely changes what remains.