By · Updated

Binomial Probability Calculator

Probability of Exactly, At Most or At Least k Successes in n Trials

No ratings yet
Result

—

OneTask is free, with no sign-up, and built by one person.
If this tool saved you time, a coffee helps keep new ones coming.

How the Binomial Probability Calculator works

This binomial probability calculator works out the chance of a given number of successes in a fixed number of independent trials. Enter the trials, the probability of success on each and the number of successes, and it returns exactly, fewer than, at most, more than and at least.

A binomial setting has a fixed number of trials, each with two outcomes and the same chance of success, and trials that do not affect one another. Coin flips, free throws, defective parts on a production line and yes-or-no survey answers all fit, as long as those conditions hold.

How the binomial probability calculator works

The formula is P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, where C(n, k) counts the ways to choose which k trials succeed. Cumulative answers such as P(X ≤ k) add these terms from 0 up to k. The mean is n × p and the standard deviation √(n × p × (1 − p)).

First example. The chance of exactly 7 heads in 10 fair flips is C(10, 7) × 0.5¹⁰ = 120 ÷ 1024 ≈ 0.117.

Second example. A 70% free-throw shooter takes 10 shots. The chance of at least 8 is about 0.383.

How many successes should you expect? On average n × p, so 12 in 20 trials at 60%.

Binomial distribution for n = 20 and p = 0.5 around the mean

Successes k P(X = k) P(X ≤ k) P(X ≥ k)
0 0.000001 0.000001 1.000000
1 0.000019 0.000020 0.999999
2 0.000181 0.000201 0.999980
3 0.001087 0.001288 0.999799
4 0.004621 0.005909 0.998712
5 0.014786 0.020695 0.994091
6 0.036964 0.057659 0.979305
7 0.073929 0.131588 0.942341
8 0.120134 0.251722 0.868412
9 0.160179 0.411901 0.748278
10 0.176197 0.588099 0.588099
11 0.160179 0.748278 0.411901
12 0.120134 0.868412 0.251722
13 0.073929 0.942341 0.131588
14 0.036964 0.979305 0.057659
15 0.014786 0.994091 0.020695
16 0.004621 0.998712 0.005909
17 0.001087 0.999799 0.001288
18 0.000181 0.999980 0.000201
19 0.000019 0.999999 0.000020
20 0.000001 1.000000 0.000001

Tips

FAQ

What is the difference between P(X ≤ k) and P(X < k)?

P(X ≤ k) includes k itself, while P(X < k) stops one short. For "at most 3" use ≤ 3; for "fewer than 3" use < 3, which is the same as ≤ 2. With whole-number counts the distinction always matters, unlike a continuous distribution where single points have zero probability.

When can I use the normal approximation to the binomial?

When both n × p and n × (1 − p) are at least about 10, the binomial is close to a normal curve with the same mean and standard deviation. This calculator computes the exact binomial probabilities, so you only need the approximation for hand working or for very large n.

What does a binomial table show?

It lists P(X = k) and the cumulative P(X ≤ k) for each k at a fixed n and p. The table on this page updates with your inputs, so you can scan the whole distribution and see where the probability is concentrated rather than looking up one row at a time.

See also: all Statistics tools, Arithmetic, Number Theory, Algebra and Geometry, Trigonometry and Calculus.

Related tools

Probability Calculator

Combine two probabilities, and the odds of at least once.

Permutation And Combination Calculator

nCr and nPr, with and without repetition, exact to the last digit.

Normal Distribution Calculator

Areas under any bell curve, and values at a percentile.

Expected Value Calculator

Expected value, variance and SD of a set of outcomes.

Average Calculator

Mean, median, mode and range of any list of numbers.