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Mathematics · Glossary

What is Binomial distribution?

Definition 33.7 High School Mathematics · Chapter 33 — Random Variables and the Binomial Distribution

Repeat a Bernoulli trial nn times independently, and let XX be the total number of successes. The distribution of XX is the binomial distribution B(n,p)\mathcal B(n, p).

Three Bernoulli trials: exactly 32 = 3 of the 23 paths give two successes (red), each with probability p2(1-p), so (X = 2) = 3p2(1-p).
Three Bernoulli trials: exactly (32)=3\binom{3}{2} = 3 of the 232^3 paths give two successes (red), each with probability p2(1−p)p^2(1-p), so P(X=2)=3p2(1−p)\P(X = 2) = 3p^2(1-p).
The distribution B(20, 0.3): mean np = 6, standard deviation √npq 2.05.
The distribution B(20,0.3)\mathcal B(20, 0.3): mean np=6np = 6, standard deviation npq≈2.05\sqrt{npq} \approx 2.05.
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