Probability calculator

Probability is favourable outcomes over total outcomes. For independent events, both happening is the product; at least one is 1 minus the product of the complements. Over n tries at probability p, at least once is 1 − (1−p)ⁿ.

Independence is the assumption that quietly breaks these formulas. Two dice are independent; two cards drawn without replacement are not, because the first draw changes what is left. Rainfall on consecutive days is not independent either. Where events are dependent, the multiplication rule needs conditional probabilities instead — P(A and B) = P(A) × P(B given A) — and using the simple product will overstate or understate depending on which way the dependence runs.