Binomial Distribution Calculator
Exact binomial probabilities for “number of successes in n trials” — with the full PMF chart.
Needs fixed n, two outcomes, independent trials, constant p. Example: 10 fair coin flips, P(X = 3).
① Answer
② Steps
③ Why it works
The binomial PMF counts paths: C(n,k) ways to choose which k trials succeed, each path having probability pk(1−p)n−k. Cumulative probabilities just add PMF bars — “at least” and “at most” are complements, which is why P(X ≥ k) = 1 − P(X ≤ k−1). The mean np sits at the distribution’s balance point.