Probability Calculator
Work out the probability of two events, or the chance of exactly k successes in repeated trials (binomial).
The basics
Probability measures how likely something is, from 0% (impossible) to 100% (certain). For two events, the formulas depend on how they relate: independent events do not affect each other, such as two separate coin flips. Mutually exclusive events cannot both happen, such as rolling a 2 and a 5 with one die.
Repeated trials
If each trial has the same chance of success and the trials are independent, the number of successes follows the binomial distribution, which the second tab uses.
Example
On the first tab (two events) with the starting values shown (Probability of A 30 %; Probability of B 40 %; The events are Independent), A or B comes out as 58%. Other results include a and b both happen: 12%; neither happens: 42%; exactly one happens: 46%.
The binomial probability formula
If you repeat an experiment n times, each with the same chance p of success, the probability of exactly k successes is P(X = k) = C(n, k) × pk × (1 − p)n − k, where C(n, k) is the number of ways to choose which k trials succeed. The expected number of successes is n × p.
Examples
| Question | Answer |
|---|---|
| Exactly 5 heads in 10 coin flips | 24.61% |
| At least one 6 in 4 rolls of a die | 51.77% |
| Exactly 3 successes in 20 trials at 10% each | 19.01% |
| Exactly 2 defective items in 10 when 5% are defective | 7.46% |
Two events
For independent events, P(A and B) = P(A) × P(B), and P(A or B) = P(A) + P(B) − P(A and B). For example, with P(A) = 30% and P(B) = 40%, the chance both happen is 12%, and the chance that at least one happens is 58%.
Cumulative probabilities
To find "at most k" or "between a and b" successes, add the exact probabilities for each value. For "at least one", use 1 minus the chance of none, which is the quickest method.
Next steps: Permutation and Combination Calculator, Z-Score Calculator, Standard Deviation Calculator.
Frequently asked questions
What is a binomial distribution?
It describes the number of successes in a fixed number of independent trials, each with the same probability of success.
How do I calculate the probability of at least one success?
Calculate 1 − (1 − p)^n, where p is the chance of success in each trial and n is the number of trials.
When can I use the binomial formula?
When there is a fixed number of trials, each trial has two outcomes, the chance of success is the same each time and the trials are independent.
What is the difference between independent and dependent events?
Independent events do not affect each other, like coin flips. With dependent events, one outcome changes the odds of the other, like drawing cards without replacement.
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