Binomial Distribution Calculator

How to use

1. 1Pick a region: P(X = k) for an exact count, P(X <= k) or P(X < k) for a left tail, P(X >= k) or P(X > k) for a right tail, or P(a <= X <= b) for a range.
2. 2Enter n, the number of independent trials (whole number from 0 to 100,000), and p, the per-trial success probability. p accepts decimals (0.5), percents (50%), or fractions (1/6).
3. 3Enter k for a single-bound region, or both a and b for the range region. All counts must be whole numbers between 0 and n.
4. 4Read the probability as a decimal and a percent, plus the mean, variance, standard deviation, mode, skewness, and excess kurtosis. The PMF bar chart shades the bars inside your chosen region.
5. 5Use the example chips to load classic textbook scenarios in one click, or Copy summary or Copy full report to paste the numbers into homework, a spreadsheet, or a launch doc.

About this tool

Binomial Distribution Calculator returns the probability of seeing a given number of successes in n independent trials with constant success probability p, plus every statistic students, teachers, A/B test analysts, and quality engineers actually want next to the answer. Pick one of six probability regions in a single click: P(X = k) for an exact count (the textbook binompdf), P(X <= k) and P(X < k) for left-tail cumulative probabilities (binomcdf), P(X >= k) and P(X > k) for right-tail probabilities, and P(a <= X <= b) for a range that is closed on both ends. Enter n as a whole number (up to 100,000), enter p as a decimal (0.25), a percent (25%), or a fraction (1/6), and the result panel updates instantly with the probability to six significant figures and to four decimal places of percent, the mean np, the variance np(1 - p), the standard deviation, the mode floor((n + 1) p), the skewness (1 - 2p) / sqrt(np(1 - p)), and the excess kurtosis (1 - 6p(1 - p)) / (np(1 - p)). A live SVG bar chart of the PMF shades the bars inside the requested region in blue and the rest in slate, so the answer is visible at a glance; for large n the chart automatically zooms to the most probable bars around the mode so the visualization stays readable. Numerically, the PMF is evaluated in log space using a Lanczos log-gamma approximation for the log-binomial-coefficient, and the CDF is summed outward from the mode so the dominant terms anchor the running total. This stays stable and precise for n up to 100,000 with sub-millisecond evaluation in modern browsers, which the naive C(n, k) p^k (1 - p)^(n - k) formula cannot do (it overflows past n around 1000). Useful for high-school and college statistics homework, AP Statistics binompdf and binomcdf questions, A/B test sanity checks before opening a full significance calculator, sample size and acceptance sampling problems in quality control, defect-rate and reliability work, finance scenario analysis (number of in-the-money options, number of defaults), epidemiology and medical screening intuition (sensitivity and specificity), poker and dice probability problems, and any time you have a count out of a fixed number of trials with a constant per-trial probability. Quick example chips load common textbook scenarios (10 coin flips, 20 free throws at 70%, 50 dice rolls at 1/6, 100 visitors at 8% conversion, 1000 emails at a 0.2% spam rate) so you can verify a class problem in a single click. Everything runs locally in your browser, so the numbers you type for class, work, or research never leave your device.

Free to use. Works in your browser. No signup, no login.