Probability Calculator
The probability of two events together or either, a conditional probability by Bayes’ theorem, or k successes in n independent trials.
Result
Probability: 15%
Exact arithmetic on the probabilities entered. Nothing is estimated.
- Which probability
- P(A and B), for independent events
- P(A or B)
- 65%
- P(not A)
- 50%
- P(not B)
- 70%
- P(neither)
- 35%
https://calcgallery.com/statistics/probability-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Probability Calculator
- Choose the question. Two events together or either, one event given another, or a number of successes in repeated tries.
- Enter the probabilities as percentages, and for two events say whether they are independent or mutually exclusive.
- Read the probability, with the related ones below it.
The rules
| Question | Formula |
|---|---|
| Not A | 1 − P(A) |
| A and B, independent | P(A) × P(B) |
| A or B, independent | P(A) + P(B) − P(A) × P(B) |
| A and B, mutually exclusive | 0 |
| A or B, mutually exclusive | P(A) + P(B) |
| A given B | P(B given A) × P(A) ÷ P(B) |
| At least one success in n tries | 1 − (1 − p)ⁿ |
| Exactly k successes in n tries | C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ |
For “or”, the overlap is subtracted once because it would otherwise be counted in both P(A) and P(B). Mutually exclusive events have no overlap, so there is nothing to subtract.
Repeated trials
When the same event is tried n times, each try independent of the others and with the same chance p, the number of successes follows the binomial distribution. The page gives the chance of exactly k successes, at least k, at most k, at least one and none. C(n, k) counts the different orders the k successes could come in; the permutation and combination calculator explains it.
The page works the binomial term out in logarithms, so a thousand trials is as reliable as ten. Worked out directly, C(1000, 500) is too large for a computer’s ordinary numbers.
Probabilities as percentages
Every probability here is entered and shown as a percentage, from 0% (impossible) to 100% (certain). A probability of 0.3 is 30%.
With your numbers
- P(A or B) = 50% + 30% − 50% × 30% = 65%
- P(A and B) = 50% × 30% = 15%
Quick reference
Exact arithmetic on the probabilities entered. Nothing is estimated.
| Example | Probability |
|---|---|
| 50% and 30%, independent | 15% |
| 50% and 30%, mutually exclusive | 0% |
| Two coin flips both heads | 25% |
| 1% condition, 90% detection, 5% positive | 18% |
| At least one in 10 tries at 10% | 38.742% |
| Exactly 5 heads in 10 flips | 24.6094% |
Common questions about the Probability Calculator
- What is the difference between independent and mutually exclusive?
- Independent events do not affect each other: a coin landing heads says nothing about the next roll of a die. Mutually exclusive events cannot happen together: one roll of a die cannot be both a 2 and a 5. They are opposites. If two events with any chance of happening are mutually exclusive, knowing one happened tells you the other did not, so they cannot be independent.
- Why is a positive test not proof of the condition?
- Because most positive results come from the much larger group without it. In the screening example on this page, 1% of people have a condition, the test catches 90% of them, and 5% of all tests come back positive. Only 18% of positives actually have the condition. That is Bayes’ theorem, and it is why a rare condition needs a second test to confirm it.
- What does "at least one" mean over repeated tries?
- The chance that something with probability p happens at least once in n independent tries is 1 − (1 − p)ⁿ. It is easier to work out the chance it never happens and subtract that from 1. A 10% chance each time gives about a 65% chance of at least one success in ten tries, not 100%.
- Why did the page refuse my numbers?
- Because they cannot all be true at once. Two mutually exclusive events cannot have probabilities adding up to more than 100%. P(B given A) × P(A) is the chance of both A and B, which cannot be more than the chance of B alone. When inputs contradict each other, the page gives no result rather than a probability over 100%.