Binomial Probability Calculator

What this calculator does

The binomial probability calculator evaluates the binomial distribution for n independent trials each with success probability p. It computes the exact probability for four event modes: exactly k successes, at least k successes (upper cumulative tail), at most k successes (lower cumulative tail), and a range from k to a specified end value. Alongside the requested probability, the tool shows the full probability mass function across a configurable window of outcomes, expected value (μ = np), standard deviation (σ = √(np(1−p))), and a normal approximation with an accuracy indicator.

The binomial model

The binomial distribution models a fixed number n of independent trials, each with exactly two outcomes (success / failure) and a constant success probability p. It answers questions of the form: given n independent attempts each with probability p of success, what is the probability of observing exactly k (or at least k, or at most k) successes? The key assumptions are independence between trials and a constant p throughout: violations of either assumption make the binomial a poor fit.

Formula breakdown

• C(n,k) is the binomial coefficient: the number of distinct ways k successes can occur in n trials.
• Cumulative P(X ≥ k) sums exact probabilities from k to n.
• Cumulative P(X ≤ k) sums exact probabilities from 0 to k.
• The normal approximation μ ± z·σ is reliable when np ≥ 5 and n(1−p) ≥ 5.

Interpreting results

• A probability near 0 means the event is very unlikely in a single experiment with n trials. To interpret how often it would occur over many repeated experiments, consider its reciprocal.
• The probability mass function window shows which outcome values carry the most probability weight. The peak of the distribution falls near the expected value np.
• The normal approximation accuracy label tells you whether the approximation is reliable for your n and p. When it is flagged as weak, rely only on the exact binomial result.
• The percentile threshold field lets you find the smallest k such that P(X ≤ k) exceeds a given percentile: useful for setting confidence thresholds or risk limits.

Real-world scenarios

• Quality control: with a 2% defect rate and a batch of 100 items, what is the probability of finding 5 or more defects? Use at-least mode with n=100, p=0.02, k=5.
• A/B testing: if a control variant converts at 10% and you run 50 trials, what is the probability the treatment variant sees 8 or more conversions just by chance? Use at-least mode to set significance boundaries.
• Sports analytics: a basketball player makes 75% of free throws. In 10 attempts, what is the exact probability of making exactly 8? Use exactly mode with n=10, p=0.75, k=8.
• Software testing: if each test case has a 5% chance of finding a bug, and you run 20 tests, what is the probability of finding at least one bug? Use at-least mode with k=1.

Edge cases

• p = 0: P(X=0) = 1, all other probabilities are 0. No successes will ever occur.
• p = 1: P(X=n) = 1, all other probabilities are 0. Every trial will succeed.
• k > n: impossible event: probability is 0.
• Very large n (hundreds or thousands): exact computation uses log-space arithmetic to avoid overflow. Results remain numerically stable.
• Normal approximation when np < 5 or n(1−p) < 5: the tool flags this and the approximation should not be used.