About the Probability Calculator
This probability calculator combines two events, A and B. Enter the chance of each as a percentage, say whether they are independent (one does not affect the other, like two coin flips) or mutually exclusive (they cannot both happen, like rolling a 1 and a 6 on one die), and it returns the probability of both, either, exactly one, neither, and the complement of each.
It also answers the repeated-trials question people often get wrong: if something has a 5% chance each time, what is the chance it happens at least once in 20 tries? The answer — about 64%, not 100% — comes from multiplying the chances of it not happening. You also get the expected number of successes and the odds of A.
Probabilities here are entered as percentages (50 for a one-in-two chance). Results assume the rule you choose; if the events are dependent in some other way, use P(A and B) = P(A) × P(B | A) with the conditional probability you know.
With the default inputs, the p(a and b) — both happen is 18%. Change any value above to recalculate instantly.
How to use the probability calculator
- 1Enter the probability of event A as a percentage.
- 2Enter the probability of event B.
- 3Choose whether the events are independent or mutually exclusive.
- 4Set the number of repeated trials to see the chance of A happening at least once.
- 5Read P(A and B), P(A or B) and the other combinations, and compare them on the chart.
Formula and method
For independent events the chance that both happen is the product of their probabilities. The chance that at least one happens follows the addition rule, subtracting the overlap so it is not counted twice. For mutually exclusive events the overlap is zero, so P(A and B) = 0 and P(A or B) is simply P(A) + P(B).
The complement rule, P(not A) = 1 − P(A), gives "neither" as 1 − P(A or B) and "exactly one" as P(A or B) − P(A and B). For repeated independent trials, the chance of never seeing A in n tries is (1 − P(A))ⁿ, so the chance of at least one success is one minus that, and the expected count is n × P(A).
- P(A), P(B)
- Probability of each event, 0 to 1 (entered as %)
- A∩B
- Both A and B happen
- A∪B
- A or B or both happen
- n
- Number of independent repeated trials
Worked examples
Two independent events, 60% and 30%
Both: 0.6 × 0.3 = 0.18 (18%). Either: 0.6 + 0.3 − 0.18 = 0.72. Neither: 1 − 0.72 = 28%. Over 10 trials, A fails every time with probability 0.4¹⁰ ≈ 0.0001, so it happens at least once 99.99% of the time.
Rolling a 1 or a 6 on one die (mutually exclusive)
One die cannot show both numbers, so P(both) = 0 and P(1 or 6) = 1/6 + 1/6 = 1/3. The chance of rolling at least one 1 in six rolls is 1 − (5/6)⁶ ≈ 66.5%, not 100%.
A 5% chance tried 20 times
Each attempt fails 95% of the time, so all 20 fail with probability 0.95²⁰ ≈ 0.358. The chance of at least one success is therefore about 64.2%, and on average you would expect one success.
Frequently asked questions
How do you calculate the probability of two independent events?+
Multiply them. If P(A) = 0.5 and P(B) = 0.2 and the events do not influence each other, the probability that both happen is 0.5 × 0.2 = 0.1, or 10%.
What is the difference between independent and mutually exclusive events?+
Independent events do not affect each other’s chances and can happen together. Mutually exclusive events cannot happen at the same time, so knowing one occurred tells you the other did not — they are actually strongly dependent.
How do I find the probability of A or B?+
Use the addition rule: P(A or B) = P(A) + P(B) − P(A and B). Subtracting the overlap avoids double counting. For mutually exclusive events the overlap is zero, so just add them.
What is the probability of something happening at least once?+
Take 1 minus the probability it never happens. With a 10% chance per try over 10 tries: 1 − 0.9¹⁰ ≈ 65.1%. Adding 10% ten times to get 100% is a common mistake.
How do I convert probability to odds?+
Odds in favour are P ÷ (1 − P). A 60% probability is 0.6 ÷ 0.4 = 1.5, or 3 : 2 in favour. Odds against are the reverse, 2 : 3.