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Binomial Distribution Calculator
The binomial distribution models the number of successes in n independent trials that each succeed with probability p: correct guesses on a multiple-choice test, defective items in a batch, heads in a series of coin flips. Enter n and p, choose the probability you need and the value(s) of k, and the calculator evaluates the formula P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ and sums the required terms.
The result panel shows the single-term calculation with the actual numbers, the cumulative rule used (for example P(X ≥ k) = 1 − P(X ≤ k − 1)), the mean np, variance np(1 − p) and standard deviation, a bar chart with the selected values highlighted, and a table of P(X = k), P(X ≤ k) and P(X ≥ k) for every k.
This matches binompdf and binomcdf on the TI-84 and BINOM.DIST in Excel. For rare events with large n, the Poisson distribution is a close approximation, and for large np the normal distribution with continuity correction can be used.
Result and step-by-step solution
Mean np = 3.6, variance np(1 − p) = 2.52, SD = 1.5875
- FormulaP(X = k) = C(n, k) × pk × (1 − p)n − k
- Single term at k = 4P(X = 4) = C(12, 4) × 0.34 × 0.78 = 495 × 0.0081 × 0.057648 = 0.23114
- Add the terms in the rangeP(X ≤ 4) = 0.72366
- Probability table
k P(X = k) P(X ≤ k) P(X ≥ k) 0 0.01384 0.01384 1 1 0.07118 0.08503 0.98616 2 0.16779 0.25282 0.91497 3 0.2397 0.49252 0.74718 4 0.23114 0.72366 0.50748 5 0.1585 0.88215 0.27634 6 0.07925 0.9614 0.11785 7 0.02911 0.99051 0.0386 8 0.0078 0.99831 0.00949 9 0.00149 0.99979 0.00169 10 0.00019 0.99998 0.00021 11 0.00001 1 0.00002
Frequently asked questions
- What is the difference between P(X ≤ k) and P(X < k)?
- For a discrete variable P(X < k) = P(X ≤ k − 1). The calculator applies this automatically, so choose whichever wording your question uses.
- How do I compute "at least k"?
- P(X ≥ k) = 1 − P(X ≤ k − 1). The tool shows this subtraction in the steps.
- What are the conditions for a binomial distribution?
- A fixed number of trials, each with only two outcomes, the same probability of success on every trial, and independence between trials.
- How large can n be?
- Up to 100,000. The calculator uses logarithms of the binomial coefficient, so large n does not overflow.
Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.