Probability Calculator
Calculate a single-event probability, combine two events, find a union with a known intersection, or solve a binomial probability.
Probability formulas
For equally likely outcomes, P(A) = favorable outcomes ÷ total outcomes, and the complement is P(not A) = 1 − P(A).
For independent events, P(A and B) = P(A) × P(B). The probability of at least one is 1 − [1 − P(A)] × [1 − P(B)].
When the intersection is known, P(A or B) = P(A) + P(B) − P(A and B). Do not use this as an independence formula: the intersection must be known or justified separately.
For n independent trials with constant success probability p, the binomial probability of exactly k successes is C(n,k) × pk × (1 − p)n−k.
Checked examples
| Calculation | Inputs | Result |
|---|---|---|
| Single event | 1 favorable out of 6 | 16.6666666667% |
| Single event | 0 favorable out of 10 | 0% |
| Single event | 10 favorable out of 10 | 100% |
| Independent events | P(A)=50%, P(B)=50% | Both 25%; at least one 75% |
| Independent events | P(A)=20%, P(B)=30% | Both 6%; exactly one 38% |
| Union | P(A)=50%, P(B)=40%, both=20% | A or B = 70% |
| Binomial | n=10, k=3, p=50% | Exactly 11.71875% |
| Binomial | n=4, k=2, p=50% | Exactly 37.5%; at most 68.75%; at least 68.75% |
How to use the Probability Calculator
- Select the calculation type that matches the information you know.
- Enter counts as whole numbers or probabilities as percentages from 0 to 100.
- Select Calculate probability.
- Review the primary result and its complement or related probabilities.
Choosing the correct mode
- Favorable and total outcomes: use only when the possible outcomes are equally likely.
- Two independent events: use when the occurrence of one event does not change the probability of the other.
- Union: use when P(A), P(B) and their overlap P(A and B) are known.
- Binomial: use for a fixed number of independent trials, two outcomes per trial and a constant success probability.
Limits and interpretation
- Probabilities must be from 0% to 100%.
- Outcome counts must be whole numbers; favorable outcomes cannot exceed total outcomes.
- The union mode checks whether the supplied intersection is mathematically possible.
- The binomial mode supports 1–5,000 trials and requires successes from 0 through n.
- Very small probabilities may display in scientific notation. Calculations use unrounded values.
- A valid calculation does not prove that real-world events are equally likely, independent or binomial. Those assumptions come from the problem context.
Frequently asked questions
What is the difference between independent and mutually exclusive events?
Independent events can occur together and do not change each other's probability. Mutually exclusive events cannot occur together, so their intersection is zero.
Why is the intersection subtracted in the union formula?
Adding P(A) and P(B) counts the overlap twice. Subtracting P(A and B) once corrects that double count.
What does “at least k” include?
It includes k successes and every larger number of successes through n. “At most k” includes zero through k.
Can I enter decimal probabilities?
The inputs use percentages. Enter 12.5 for 12.5%, not 0.125.
Related calculators
Use the Permutation and Combination Calculator for nPr, nCr and arrangements, the Random Number Generator for random samples, or the Standard Deviation Calculator for data spread and z-scores.