Free Binomial Probability Calculator
Calculate the probability of exactly k successes in n independent Bernoulli trials with success probability p.
P(X = k)
0.11718750
Expected Value (np)5.0000
Standard Deviation1.5811
P(X = k) vs Number of Trials (n)
How to Calculate Binomial Probability
Formula
P(X = k) = C(n,k) * p^k * (1-p)^(n-k)
The binomial distribution models the number of successes in n independent trials, each with the same probability p. C(n,k) counts the number of ways to arrange k successes among n trials. The mean is np and the variance is np(1-p).
Example Calculation
Flip a fair coin 10 times. What is the probability of exactly 3 heads?
- 01C(10,3) = 10! / (3! * 7!) = 120
- 02p^k = 0.5^3 = 0.125
- 03(1-p)^(n-k) = 0.5^7 = 0.0078125
- 04P(X=3) = 120 * 0.125 * 0.0078125 = 0.1171875
- 05Expected value = 10 * 0.5 = 5
Frequently Asked Questions
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