Probability Calculator (Binomial)

Binomial Probability - Quick Cheat Sheet

Distribution: X ~ Binomial(n, p), where n = # of independent trials, p = probability of success each trial.
PMF (exact): P(X = k) = C(n,k) · p^k · (1-p)^n-k.
At most: P(X ≤ k) = Σ_i=0..k P(X = i).
At least: P(X ≥ k) = 1 - P(X ≤ k - 1).
Between: P(a ≤ X ≤ b) = Σ_i=a..b P(X = i).
Mean = n·p; Variance = n·p·(1-p); sd = √(n·p·(1-p)).
Inputs: k must be 0..n, p in [0, 1]. For "Between", ensure 0 ≤ a ≤ b ≤ n (the tool will clamp to range).
Tip: For very large n, exact summation can be slow; consider a normal or Poisson approximation offline if needed.

Edge cases & notes

If k=0 in "at least", result is 1 (since P(X ≥ 0) = 1).
If a>b is entered for "Between", the calculator automatically swaps them.
All calculations are done using PHP floating-point arithmetic; extremely large n may lose precision.