Distributions
Probability Calculator
Single events, combined events, and the chance that something happens at least once across many tries.
How to read this result
Probability is a proportion of outcomes when every outcome is equally likely — the classical definition, and the one the first mode uses. That assumption is doing real work: it holds for a fair die and a shuffled deck, and not for "it either rains or it does not".
"And" multiplies, "or" adds — with a correction. P(A and B) = P(A)P(B) for independent events. P(A or B) = P(A) + P(B) − P(A and B), and the subtraction is the step people skip: without it the overlap is counted twice and the answer can exceed 1.
Independent and mutually exclusive are opposites, not synonyms. Mutually exclusive events cannot both happen, so learning one occurred tells you the other did not — which is maximal dependence, the very opposite of independence.
For "at least one", compute the complement. The probability that something happens at least once in n tries is 1 minus the probability it never happens: 1 − (1 − p)ⁿ. This is why rare events become near-certain at scale, and why "one in a million" happens constantly to a large enough population.
The formula
The addition rule. For independent events P(A ∩ B) = P(A)P(B); otherwise P(A ∩ B) = P(A)·P(B | A).
At least one occurrence
Computed with log1p and expm1 rather than directly, so a tiny p across many trials keeps its precision instead of rounding to zero.
Worked example
At least one six in four rolls
A classic gambling problem — the one that started probability theory. What is the chance of rolling at least one six in four throws of a fair die?
- The chance of a six on one roll.1/6 = 0.16667
- The chance of no six on one roll.5/6 = 0.83333
- Four independent rolls: multiply.(5/6)⁴ = 0.48225
- At least one six is the complement.1 − 0.48225 = 0.51775
Just over half — which made this a profitable bet for the seventeenth-century gambler who noticed it. The same arithmetic on 24 rolls of two dice for a double six gives 49.1%, a losing bet, and the discrepancy is what prompted Pascal and Fermat's correspondence.
Checked against 1 − (5/6)⁴, computed exactly.
The calculator above is loaded with these numbers by the Load the worked example button.
Assumptions, and when to use something else
The classical definition assumes every listed outcome is equally likely. The multiplication rule assumes independence unless you supply a conditional probability. The "at least one" formula assumes n independent trials with a constant p.
Where these break in practice is dependence people do not notice: two components from the same production batch, two claims from the same storm, two customers who heard the same advertisement. Assuming independence when it does not hold understates the probability of everything failing at once, sometimes by orders of magnitude.
- Counting successes in a fixed number of trialsBinomial distributionThe full distribution, not just “at least one”.
- Counting arrangementsPermutations and combinationsHow many ways there are, which is usually the denominator.
- Updating a probability with evidenceBayes' theoremConditional probability run backwards.
- Averaging outcomes with different valuesExpected valueWhat a random quantity is worth in the long run.
Questions people ask
What is the difference between independent and mutually exclusive events?
Independent means one has no bearing on the other, so P(A and B) = P(A)P(B). Mutually exclusive means they cannot both happen, so P(A and B) = 0. Mutually exclusive events with non-zero probabilities are maximally dependent, since knowing one occurred tells you the other did not.
How do I calculate the probability of at least one?
Take the complement. The probability of at least one occurrence in n independent tries is 1 − (1 − p)ⁿ. Computing it directly by adding the cases for exactly one, exactly two, and so on gives the same answer with far more work.
What does conditional probability mean?
P(B | A) is the probability of B given that A has happened — the probability restricted to the world where A is true. It is the building block of Bayes' theorem and the reason dependence has to be handled explicitly.
Can a probability be greater than 1?
No. If a calculation produces one, the usual cause is adding probabilities of events that can happen together without subtracting the overlap.