Binomial Distribution Calculator

Binomial Distribution Formula

Worked Example

3 heads in 10 tosses of a fair coin:

1. P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ
2. = C(10, 3) × 0.5^3 × 0.5^7
3. = 0.1171875

P(X < 3): 0.0546875 · P(X ≥ 3): 0.9453125 · P(X > 3): 0.828125 · Mean np: 5 · Variance npq: 2.5 · Standard deviation √(npq): 1.58114

Exactly, at most and at least

GATE problems often ask for “at least one” success, which is easiest as 1 − P(X = 0) = 1 − (1 − p)ⁿ. “At most k” is the cumulative P(X ≤ k), and “more than k” is 1 − P(X ≤ k). The trials must be independent with the same p each time; sampling without replacement from a small batch is hypergeometric instead. For large n the normal approximation with mean np and variance np(1 − p) can be used.

More Engineering Mathematics Calculators

• Poisson Distribution Calculator

  P(X = k) = e^(−λ) λᵏ ÷ k!

  λ mean number of events in the interval; mean = variance = λ

  Example: Exactly 3 events when 2 are expected (λ = 2): P(X = 3) = 0.18044704.

  Open the poisson distribution calculator →

• Eigenvalue Calculator

  det(A − λI) = 0

  Σλ = trace A, Πλ = det A

  Example: The matrix [[4, 1], [2, 3]]: Eigenvalues: λ₁ = 5, λ₂ = 2.

  Open the eigenvalue calculator →

• Simpson’s Rule Calculator

  I = h/3 [y₀ + 4(y₁ + y₃ + …) + 2(y₂ + y₄ + …) + yₙ]

  h step size, n even number of intervals

  Example: y = 1/(1 + x²) at x = 0, 0.25, …, 1 (h = 0.25): Integral I ≈ 0.785392.

  Open the simpson’s rule calculator →

Frequently asked questions

Sources

Last reviewed: 2026-10-15. Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.