Binomial CDF calculator

The binomial cumulative distribution function is F(k) = P(X≤k). It includes k as well as every smaller possible success count. For an integer count, “at most k,” “no more than k” and “k or fewer” mean the same event.

F(k) = ∑_j=0^k C(n,j) p^j (1−p)^n−j

Each term is a single-count PMF probability. The CDF adds them. As k increases, the CDF never decreases, and F(n) = 1.

This separate example uses n = 4, p = 0.5 and k = 2. There are 16 equally likely success/failure arrangements under those assumptions.

P(X≤2) = (1 + 4 + 6) / 16 = 0.6875 = 68.75%. Exactly two would be 6/16, so using the PMF alone would leave out zero and one success.

• Fewer than k excludes k. P(X<k) = F(k−1).
• More than k excludes k. P(X>k) = 1−F(k).
• At least k includes k. P(X≥k) = 1−F(k−1).

For boundary reasoning, define F(−1) = 0. Then at least zero has probability 1. The form only accepts k from 0 through n; use its comparison choice rather than entering −1.

The distinction between 1−F(k) and 1−F(k−1) is one count. The at least vs more than guide works through both with the same inputs.

These identities explain the math. For tiny tails, subtracting a rounded CDF from 1 can erase a real probability. The calculator sums the selected event and its complement separately.

The inverse question starts with a cumulative target instead of a known k. With n = 4 and p = 0.5, a target of 0.8 gives k = 3. F(2) = 0.6875 is too small, while F(3) = 0.9375 reaches the target.

There may be no count with a CDF exactly equal to your target because the distribution moves in steps. This tool returns the smallest supported count that reaches or exceeds it. If the target equals F(k) exactly, the result stays at k.

This is a lower-tail quantile. It does not find a minimum trial count, estimate the success probability, or solve a hypothesis test. If you need a single event probability, use the calculator at the top of this page.

After a forward calculation, open the full probability table and switch the graph to CDF. Each table row includes both PMF and CDF; the step chart shows how cumulative mass grows. All n + 1 possible counts are included.

Use independent, two-outcome trials with a constant p. The forward tool supports n from 0 to 1000 and preserves tiny probabilities; the inverse tool states its separate decimal-place limits above. Values shown in the table are rounded.

For choosing parameters from an exercise, see how to calculate binomial probability. The formula and discrete quantile definition are described in NIST’s binomial reference.