Enter probabilities as decimals (0.25) or percentages (25%).
About this probability calculator
Probability measures how likely an event is, from 0 (impossible) to 1 (certain). This calculator handles the three situations people search for most. For two events A and B it finds the chance of both, either, neither and one given the other. For repeated independent trials it gives the exact binomial probability of exactly, at most or at least k successes. And for a single chance repeated several times it shows the probability of at least one success. Every result is shown as a decimal, a percentage and where useful as a fraction.
Key formulas
| Question | Formula |
|---|---|
| Not A (complement) | P(A′) = 1 − P(A) |
| A and B, independent | P(A ∩ B) = P(A) × P(B) |
| A or B | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) |
| A and B, mutually exclusive | P(A ∩ B) = 0 |
| A given B | P(A | B) = P(A ∩ B) / P(B) |
| Binomial, exactly k | P(X = k) = C(n, k) pk (1 − p)n−k |
| At least one in n tries | 1 − (1 − p)n |
Worked examples
- Two independent events: P(A) = 0.4 and P(B) = 0.3 give P(A and B) = 0.12 and P(A or B) = 0.4 + 0.3 − 0.12 = 0.58.
- Binomial: flipping a fair coin 10 times, the chance of exactly 3 heads is C(10,3) × 0.510 = 120 / 1024 ≈ 0.1172.
- At least one: rolling a die 6 times, the chance of at least one six is 1 − (5/6)6 ≈ 0.6651, not 100%.
Independent versus mutually exclusive
These are often confused. Independent events do not affect each other, so knowing one happened tells you nothing about the other. Mutually exclusive events cannot happen together at all. If two events with non-zero probabilities are mutually exclusive, they are not independent, because one occurring makes the other impossible.
When the binomial applies
- A fixed number of trials n.
- Each trial has two outcomes, success or failure.
- The chance of success p is the same every time.
- Trials are independent.
The binomial results include the mean (np), variance (np(1 − p)) and standard deviation, plus a full probability table you can read for any k.
Related tools: summarize data with the Standard Deviation Calculator, test a result with the P-Value Calculator, or work with chance in the Random Number Generator and Percentage Calculator.
Frequently asked questions
If the events are independent, multiply their probabilities: P(A and B) = P(A) times P(B). If they are not independent, multiply P(A) by the probability of B given A. This calculator handles independent, mutually exclusive and known-intersection cases.
"And" means both events happen, which is the intersection. "Or" means at least one happens, which is the union, found by adding the probabilities and subtracting the overlap so it is not counted twice.
Binomial probability is the chance of getting exactly k successes in n independent trials when each trial has the same success probability p. It is used for coin flips, pass or fail tests, free throws and yes or no survey answers.
Take the complement: the chance of no successes in n tries is (1 minus p) to the power n, so the chance of at least one is 1 minus that value. For a 1 in 6 chance repeated 6 times it is about 66.5%.
No. A probability always lies between 0 and 1 inclusive, or 0% and 100%. If a calculation gives something outside that range, there is an error in the inputs or formula.
Odds in favour are p divided by (1 minus p). A probability of 0.25 gives odds of 0.25 / 0.75 = 1 to 3 in favour, which is 3 to 1 against.