Probability Calculator
Calculate the probability of two events happening together, either one happening, neither happening, and the chance of at least one success across repeated trials.
How to use this calculator
- 1Choose whether you are entering percentages or decimals.
- 2Enter the probability of each event.
- 3Say whether the events are independent or mutually exclusive — this changes every combined result.
- 4Set a trial count to see the "at least once" probability across repeated attempts.
How it works
Probability rules
P(A and B) = P(A) × P(B) (independent) P(A or B) = P(A) + P(B) − P(A and B) (general) P(A or B) = P(A) + P(B) (mutually exclusive) P(not A) = 1 − P(A) P(A | B) = P(A and B) ÷ P(B) P(at least one in n) = 1 − (1 − P(A))ⁿ
The addition rule subtracts the overlap because outcomes where both events occur would otherwise be counted twice. The complement rule is the most useful tool in practice: "at least one" problems are almost always easier solved as one minus the probability of none, since "none" is a single product rather than a sum over many cases.
Worked example
With P(A) = 35% and P(B) = 20% independent: both happen 7% of the time, either happens 35 + 20 − 7 = 48%, and neither happens 52%. Repeating A five times, the chance of it occurring at least once is 1 − 0.65⁵ = 88.4%.
Probability Calculator: the complete guide
Independent is not the same as mutually exclusive
These get confused constantly, and they are close to opposites. Independent means knowing one event occurred tells you nothing about the other — two coin flips. Mutually exclusive means they cannot both occur — a single die cannot land on both 3 and 5.
Mutually exclusive events are maximally dependent: if one happens, the other definitely did not. That is why P(A and B) is zero for exclusive events and P(A) × P(B) for independent ones, and why picking the wrong option above changes every figure on the page.
Why "at least once" surprises people
A 10% chance repeated ten times feels like it should be near certainty. It is 65%. Repeated twenty times it is 88%, and it never reaches 100% no matter how many trials you run — each trial multiplies the failure probability by 0.9, which approaches zero but never arrives.
The reverse surprises people too. An event with a 1-in-1,000 chance run 1,000 times occurs at least once about 63% of the time, not 100%. That figure — 1 − 1/e — is the same constant that appears in the birthday problem and in reliability engineering.
The practical lesson is that rare events become likely with enough trials, but common events never become certain.
Conditional probability and the base rate trap
P(A given B) asks how likely A is once you already know B happened. It is P(A and B) divided by P(B) — you have restricted the universe to outcomes where B occurred, so you renormalise by B's probability.
This is where medical-test intuition famously fails. A test that is 99% accurate for a disease affecting 1 in 10,000 people produces roughly 100 false positives for every true positive, so a positive result means about a 1% chance of having the disease. The test accuracy is P(positive given disease); what you want is P(disease given positive), and they are wildly different when the base rate is low.
Probability, odds, and gambling notation
Probability is successes over all outcomes. Odds are successes over failures. A 25% probability is odds of 1 to 3 — one success for every three failures — not 1 to 4. The two are converted by odds = p ÷ (1 − p).
Bookmakers quote odds against, and they include a margin, so the implied probabilities across all outcomes of an event sum to more than 100%. That excess is the house edge, and it is why the quoted odds are never the true probability.
Frequently asked questions
How do I calculate the probability of two events both happening?
For independent events, multiply them: a 35% chance and a 20% chance give 0.35 × 0.20 = 7%. For mutually exclusive events the answer is always zero, because they cannot both occur. For dependent events you need P(A) × P(B given A).
What is the probability of at least one success?
One minus the probability of no successes: 1 − (1 − p)ⁿ. For a 10% chance over 5 tries, that is 1 − 0.9⁵ = 41%. Computing it directly would require summing every case with one, two, three, four, or five successes — the complement is far simpler.
Why can't I just add probabilities together?
Because you would double-count the outcomes where both events happen. The addition rule subtracts that overlap. Straight addition is only correct for mutually exclusive events, where the overlap is zero — and it is what produces impossible answers over 100% when misapplied.
How do I convert probability to odds?
Divide the probability by its complement: odds = p ÷ (1 − p). A 20% probability is 0.2 ÷ 0.8 = 0.25, or 1 to 4 in favour — equivalently 4 to 1 against. The calculator shows the "1 in n" form, which is usually the more intuitive one.