Binomial distribution

The binomial distribution counts successes in $n$ independent trials, each with success probability $p$ . It is the backbone of coin flips, A/B tests, quality control, and opinion polls.

P (k; n, p) = (k n) p^{k} (1 - p)^{n - k}

n (trials) 20 p (success prob) 0.50 k (draws) 1000

Dark bars: PMF

P (k; n, p)

. Light bars: empirical frequencies from 1000 draws. Dashed: Normal(

n p, n p (1 - p)

) approximation.

What to notice

Symmetric at p = 0.5. The distribution is perfectly bell-shaped whenever p = 0.5. Slide p away from centre and the distribution skews toward the favoured outcome.
The Normal approximation (dashed curve) kicks in when both $n p$ and $n (1 - p)$ exceed about 5. The mean is $μ = n p$ , the standard deviation $σ = n p (1 - p)$ .
Raising n stretches the distribution and sharpens the Normal fit — the Central Limit Theorem guarantees this convergence. Try n = 50 with any p.

Relationship to Bernoulli

A single Bernoulli trial is Binomial(1, p). The binomial is the sum of n independent Bernoullis — which is exactly why the CLT applies.