Binomial Distribution Calculator
Calculate PMF, CDF, mean and variance for the binomial distribution B(n,p).
Inputs
P(X = k)
0.24609375
P(X ≤ k)
0.62304688
Mean (np)
5.0000
Variance (np(1−p))
2.5000
Step by step
Values used
Number of Trials (n) = 10; Success Probability (p) = 0.5000; Number of Successes (k) = 5
Formula applied
P(X=k) = C(n,k) × p^k × (1−p)^(n−k)
P(X = k)
= 0.24609375
P(X ≤ k)
= 0.62304688
Mean (np)
= 5.0000
Variance (np(1−p))
= 2.5000
How it works
The binomial distribution B(n, p) models the number of successes in n independent Bernoulli trials, each with success probability p. It returns the exact probability of k successes (PMF) and the cumulative probability up to k (CDF).
Formula
P(X=k) = C(n,k) × p^k × (1−p)^(n−k)
- n
- Number of trials
- k
- Number of successes
- p
- Probability of success per trial
Frequently Asked Questions
When is the binomial distribution appropriate?
When you have a fixed number of independent trials, each with the same probability of success, and you want the probability of a specific number of successes.
What is the relationship to the normal distribution?
For large n, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1−p).
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