Probability Calculator

Single Event (from counts)

Combine Independent Events

Binomial: P(X = k) where X ~ Bin(n, p)

Probability – Quick Cheat Sheet

Single event (from counts): P = successes / total, q = 1 − P.
Independence: If A and B are independent, P(A ∧ B) = P(A)·P(B), P(A ∪ B) = P(A) + P(B) − P(A)·P(B), P(¬A) = 1 − P(A).
Binomial PMF: For X ~ Bin(n, p), P(X = k) = C(n,k)·p^k·(1−p)^{n−k}.
At least one success (n trials): 1 − (1−p)^n.
Valid probability: always a number in [0, 1].
Common pitfalls: Using union formula without independence info; mixing percentages and decimals—convert to decimals (e.g., 30% → 0.30).

Edge cases & notes

Counts must satisfy 0 ≤ successes ≤ total and total ≥ 1.
Binomial inputs: n ≥ 1, 0 ≤ k ≤ n, and 0 ≤ p ≤ 1. Here n is capped at 200 for numerical stability.
“Independent” means outcome of A does not affect B; if that’s not true, the combined-event formulas shown don’t apply.