Binomial Probability Distribution Calculator (PMF, CDF & Statistics

Statistical Foundations of the Binomial Distribution

The Binomial distribution models the number of successes in a sequence of n independent experiments (Bernoulli trials):

• The Four Binomial Requirements (BINS):

• Binary: Every trial has exactly two possible outcomes (Success or Failure).

• Independent: The outcome of any trial does not affect the probability of other trials.

• Number: The total number of trials n is fixed in advance.

• Same Probability: The probability of success p remains constant for each trial.

• Combinations nk: Counts the number of distinct sequences that produce exactly k successes out of n trials.

• Normal Approximation Rule of Thumb: When np 5 and n(1-p) 5, the discrete binomial distribution is well approximated by a continuous Gaussian distribution N(np, np(1-p)) using a ± 0.5 continuity correction.

How to Use This Calculator

Enter your Number of Trials (n) (e.g. 10 (Independent repeated trials)), Probability of Success on Single Trial (p) (e.g. 0.50 (Between 0.0 and 1.0)), Target Number of Successes (k / x) (e.g. 6 (Number of successful outcomes)), and Probability Event Type to Calculate — choose from Exact Probability: P(X = k), Cumulative Probability: P(X ≤ k) [At Most k], Upper Tail Probability: P(X ≥ k) [At Least k]. Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Binomial Probability Distribution X B(n, p) equations:

Probability Mass Function (PMF): P(X = k) = nk p^k (1-p)^n-k = (n!) / (k!(n-k)!) p^k (1-p)^n-k

Cumulative Distribution (CDF): P(X k) = Σ_i=0^k ni p^i (1-p)^n-i

Upper Tail Probability: P(X k) = Σ_i=k^n ni p^i (1-p)^n-i = 1 - P(X k - 1)

Expected Value (Mean): E[X] = μ = n · p

Variance: Var(X) = σ^2 = n · p · (1-p)

Standard Deviation: σ = √(n · p · (1-p))

Worked Example

For example, flipping a fair coin) \ times and calculating the probability of getting exactly \ heads: . The exact probability is \ = 210 × (0.5)^6 × (0.5)^4 = (210) / (1024) = 0.20508 (20.51\%). The probability of getting at least 6 heads is P(X 6) = 0.37695 (37.70\%). Mean is \ and .

Understanding Your Result

Statistical results require context: a mean alone is misleading without knowing the standard deviation (spread). Normal distribution reference: 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, 99.7% within ±3 SD. For small sample sizes (n < 30), use t-distributions rather than z-scores. Statistical significance (p < 0.05) does not imply practical significance — effect size (Cohen's d, R², odds ratio) measures real-world importance. Correlation coefficient r: |r| < 0.3 weak, 0.3–0.7 moderate, > 0.7 strong.