Binomial Probability Calculator
P(X = k)0.200121
P(X ≤ k)0.849732
P(X < k)0.649611
P(X ≥ k)0.350389
P(X > k)0.150268
Mean (np)3.0000
Std Dev1.4491
The Binomial Probability Calculator models a fixed number of independent yes/no trials that each succeed with the same probability. Enter the trial count n, the target number of successes k, and the per-trial success probability p, and it returns the exact probability of exactly k successes alongside the four cumulative tails (at most, fewer than, at least, and more than k). It also reports the distribution mean np and standard deviation.
Formula
P(X = k) = C(n, k) · p^k · (1 − p)^(n − k)
- n
- Total number of independent trials
- k
- Number of successes of interest
- p
- Probability of success on a single trial
- C(n, k)
- Binomial coefficient, the count of ways to choose k from n
How it works
- Enter the number of trials n, the number of successes k you are interested in, and the single-trial success probability p between 0 and 1.
- The exact probability uses the binomial mass function P(X=k) = C(n,k)·p^k·(1−p)^(n−k), computed with logarithms of factorials so large n values stay numerically stable.
- Cumulative tails are summed from the mass function, and the summary statistics use mean = np and standard deviation = sqrt(np(1−p)).
Worked example
A process succeeds 30% of the time; in 10 trials, find P(exactly 4) and P(at most 4).
- C(10, 4) = 210, so P(X=4) = 210 · 0.3⁴ · 0.7⁶ = 210 · 0.0081 · 0.117649 = 0.200121.
- Summing P(X=0) through P(X=4) gives the cumulative P(X ≤ 4) = 0.849732.
- Mean = np = 10 · 0.3 = 3, and standard deviation = sqrt(10 · 0.3 · 0.7) = 1.4491.
P(X = 4) = 0.200121, P(X ≤ 4) = 0.849732, mean 3, SD 1.4491.
Frequently asked questions
- What conditions must hold for a binomial distribution?
- There must be a fixed number of trials, each trial independent with only two outcomes, and a constant success probability p across all trials. If p changes between trials, the binomial model no longer applies.
- Why must k be no greater than n?
- You cannot record more successes than there are trials, so k ranges from 0 to n. The calculator rejects any k larger than n as an invalid input.
- What is the difference between exact and cumulative probability?
- The exact value P(X=k) is the chance of precisely k successes. Cumulative probabilities add up a range, such as P(X ≤ k) for k or fewer successes or P(X ≥ k) for k or more, which is what most hypothesis tests need.
- How are the mean and standard deviation interpreted?
- The mean np is the expected number of successes over many repetitions, and the standard deviation sqrt(np(1−p)) measures how much the count typically varies around that expectation.