How to use the probability calculator
Enter the probability of each event as a decimal (0.25), a percent (25%) or a fraction (1/4), then say how the events are related. The calculator returns ten results at once: both, either, exactly one, neither, each event alone, each complement and both conditional probabilities, with a Venn diagram of the four regions. The second mode answers the “at least once” question for one event tried n times.
A probability is a number from 0 (impossible) to 1 (certain); the complement of A, written A′ or “not A”, has probability 1 − P(A).[1] The calculator refuses inputs that cannot be probabilities together, for example mutually exclusive events whose chances add up to more than 1.
The addition and multiplication rules
Two rules cover almost every two-event question.[3]
| You want | Rule | P(A) = 0.5, P(B) = 0.4, independent |
|---|---|---|
| P(A and B) | P(A) × P(B | A); independent: P(A) × P(B) | 0.2 |
| P(A or B) | P(A) + P(B) − P(A and B) | 0.7 |
| P(neither) | 1 − P(A or B) | 0.3 |
| P(exactly one) | P(A or B) − P(A and B) | 0.5 |
| P(A | B) | P(A and B) ÷ P(B) | 0.5 |
The minus sign in the addition rule stops the overlap from being counted twice. If the events cannot happen together (mutually exclusive), the overlap is 0 and the rule shrinks to P(A) + P(B).
Independent vs mutually exclusive events
Independent events do not affect each other: P(A | B) = P(A). Mutually exclusive events cannot both happen: P(A and B) = 0.[2] The two ideas are often confused, but they almost never hold together: if A and B both have a positive chance and are mutually exclusive, knowing B happened tells you A did not, so they are dependent.
Worked example: one card from a 52-card deck
A = a heart (13/52), B = an ace (4/52), and one card is both, the ace of hearts (1/52).
P(heart or ace) = 13/52 + 4/52 − 1/52 = 16/52 = 0.3077.
P(heart | ace) = (1/52) ÷ (4/52) = 0.25, the same as P(heart): for one card, suit and rank are independent.
Choose “I know P(A and B)” in the calculator for questions like this one, where the overlap is given rather than computed.
Probability of at least one
For one event with chance p tried n independent times, the chance it never happens is (1 − p)ⁿ, so the chance it happens at least once is 1 − (1 − p)ⁿ. A 1% chance tried 100 times is not a sure thing: 1 − 0.99¹⁰⁰ = 0.634, about 63%. The calculator also shows how many tries give an even (50%) chance.
At least one six in four rollsThe chance of at least one six in 4 rolls is 1 − (5/6)⁴ = 0.5177, just over a half: an even-money bet on it wins slightly more often than it loses. To count exactly k successes instead of “at least one”, use the binomial distribution calculator.[4]
Common probability mistakes
- Adding probabilities for “at least once”. Four rolls at 1/6 each is not 4/6; overlaps must be removed, which is what 1 − (5/6)⁴ does.
- Multiplying dependent events. Drawing two cards without replacement: the second probability changes, so use P(A) × P(B | A).
- Confusing P(A | B) with P(B | A). P(ace | heart) is 1/13, but P(heart | ace) is 1/4.
For continuous measurements use the normal distribution calculator; to judge whether a result is surprising, the p-value calculator.
How we check this calculator
Tests check exact cases: independent 0.5 and 0.4 give 0.2, 0.7, 0.3 and 0.5 for and, or, neither and exactly one; four dice give 671/1296 for at least one six; impossible combinations are rejected with a message instead of a negative probability.
- 14 November 2026: page and calculator published.
More statistics calculators
- Statistics calculatorMean, median, mode, SD, quartiles and outliers
- Standard deviation calculatorSample and population SD, step by step
- Z-score calculatorz, percentile and area under the normal curve
- P-value calculatorFrom z, t, chi-square or F
- Confidence interval calculatorMeans and proportions, t or z
- t-test calculatorOne-sample, Welch, pooled and paired
- Binomial distribution calculatorExactly, at most, at least x successes
- Normal distribution calculatorAreas, percentiles and the 68-95-99.7 rule
- Sample size calculatorSurvey size for a margin of error, and the reverse
Frequently asked questions
How do you calculate the probability of A and B?
Multiply: P(A) × P(B | A). If the events are independent this is just P(A) × P(B), for example 0.5 × 0.4 = 0.2.
How do you calculate the probability of A or B?
Add the two probabilities and subtract the overlap: P(A) + P(B) − P(A and B). For mutually exclusive events the overlap is 0.
How do you find conditional probability?
Divide the probability of both by the probability of the condition: P(A | B) = P(A and B) ÷ P(B).
What is the probability of at least one success?
1 − (1 − p)ⁿ. With a 1% chance and 100 tries it is 0.634.
Can I enter fractions?
Yes: 1/6, 0.1667 and 16.67% all work in the probability boxes.