A Bernoulli trial is a random experiment with exactly two possible outcomes: success (with probability p) and failure (with probability 1−p). Examples include flipping a coin (heads/tails), testing a product (pass/fail), or checking if a patient responds to treatment. Each trial is independent, and the probability p remains constant across all trials.

The binomial distribution models the number of successes in n independent Bernoulli trials, each with success probability p. It is defined by its probability mass function: P(X = k) = C(n,k) · p^k · (1−p)^n−k, where C(n,k) is the binomial coefficient "n choose k." The distribution is symmetric when p = 0.5 and skewed otherwise.

P(X = k) is the exact probability of getting precisely k successes. Cumulative probability (e.g., P(X ≤ k)) sums probabilities over a range of outcomes. For example, P(X ≤ 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3). This calculator supports exact, at-most, at-least, less-than, greater-than, and between-two-values cumulative probabilities.

The binomial distribution assumes: (1) Fixed number of trials n; (2) Each trial is independent; (3) Only two possible outcomes per trial (success/failure); (4) The probability of success p remains constant across all trials. If these conditions are violated, other distributions (e.g., hypergeometric for non-independent sampling without replacement) may be more appropriate.

The binomial coefficient C(n,k) = n! / (k! · (n−k)!) counts the number of ways to choose k successes from n trials. This calculator uses a numerically stable iterative method: C(n,k) = ∏_i=1^k (n−k+i)/i, which avoids computing large factorials directly and prevents overflow for n up to 1000.

For a binomial distribution with parameters n and p: Mean (μ) = n · p — this is the expected number of successes. Variance (σ²) = n · p · (1−p) — this measures the spread. The standard deviation is the square root of the variance: σ = √(n·p·(1−p)). The mode (most likely value) is typically ⌊(n+1)·p⌋ or nearby integers.

The normal approximation to the binomial is reasonable when both n·p ≥ 5 and n·(1−p) ≥ 5 (some sources use ≥10 for better accuracy). Under these conditions, the binomial distribution B(n,p) can be approximated by N(μ, σ²) with μ = n·p and σ² = n·p·(1−p). For more precise work, a continuity correction (±0.5) is recommended. For small n or extreme p values, exact binomial calculation (as this tool provides) is preferred.

Binomial probability is widely used in: Quality control (defect rates in manufacturing), Clinical trials (patient response rates), A/B testing (conversion rate comparisons), Genetics (inheritance patterns), Finance (probability of default on loans), Sports analytics (win probability in series), and Survey sampling (margin of error calculations). Any scenario with repeated independent yes/no outcomes can be modeled with binomial probability.