Probability · The basics
Probability Calculator
Combine the chances of events — both happening, either happening, one given another — and count the ways to choose or arrange things (combinations and permutations). Each answer comes with a picture of where it comes from.
Result
In plain English
Probability is just the chance of something happening, from 0 (never) to 1 (certain). The hard part is combining chances correctly — and this is where everyday intuition goes wrong most often.
- P(A and B)
- The chance both happen. For independent events you multiply; if they can't both occur (mutually exclusive) it's 0.
- P(A or B)
- The chance at least one happens. You add the two, then subtract the overlap you'd otherwise double-count.
- independent
- One event doesn't affect the other (two separate coin flips). The opposite — where one changes the odds of the other — needs conditional probability.
- P(A | B)
- “The probability of A given that B happened.” Crucially this is not the same as P(B given A) — swapping them is the classic base-rate blunder.
- at least once
- Over many tries, rare events become likely. Easiest via the back door: 1 minus the chance it never happens.
- nCr vs nPr
- Ways to pick r things from n. Combinations (nCr) ignore order (a lottery draw); permutations (nPr) count order (a race finish). nPr is always bigger.
Frequently asked
What's the difference between “and” and “or” probabilities?
For “and” (both happen) you multiply, if the events are independent. For “or” (at least one) you add the two probabilities and subtract the overlap you'd otherwise double-count. Mutually exclusive events can't both happen, so their “and” is 0 and their “or” is just the sum.
Is P(A|B) the same as P(B|A)?
No — confusing them is the base-rate fallacy. “Most people with the disease test positive” is not “most people who test positive have the disease”; the two depend on how common the condition is. Use the conditional-probability tab to see it.
What's the difference between combinations and permutations?
Combinations (nCr) count groups where order doesn't matter (a lottery draw); permutations (nPr) count arrangements where order does (a race finish). nPr is always at least as large as nCr.
What is the difference between independent and mutually exclusive events?
Independent events do not affect each other’s probability (two separate coin flips); mutually exclusive events cannot both happen at once (one die showing 3 and 5). They are often confused, but in a sense they are opposites: mutually exclusive events are strongly dependent, because knowing one happened tells you the other did not. The “and” and “or” rules differ for each, so the first step is always to decide which kind you have.