Binomial Probability Calculator Formula

Understand the math behind the binomial probability calculator. Each variable explained with a worked example.

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Formulas Used

P(X = k)

probability = coeff * pow(p, k) * pow(1 - p, n - k)

Expected Value (np)

expected_val = n * p

Standard Deviation

std_dev = sqrt(n * p * (1 - p))

Variables

VariableDescriptionDefault
nNumber of Trials (n)10
kSuccesses (k)3
pSuccess Probability (p)0.5
coeffDerived value= factorial(n) / (factorial(k) * factorial(n - k))calculated

How It Works

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).

Worked Example

Flip a fair coin 10 times. What is the probability of exactly 3 heads?

n = 10k = 3p = 0.5
  1. 01C(10,3) = 10! / (3! * 7!) = 120
  2. 02p^k = 0.5^3 = 0.125
  3. 03(1-p)^(n-k) = 0.5^7 = 0.0078125
  4. 04P(X=3) = 120 * 0.125 * 0.0078125 = 0.1171875
  5. 05Expected value = 10 * 0.5 = 5

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