Binomial Distribution

Equivalent forms

\mu=np,\quad \sigma^2=np(1-p)

The bridge distribution: counting problems in, and both the Gaussian and Poisson laws emerge in its limits.

Symbol	Name	SI unit	Dimension	Range
$P(k)$	Probabilityoutput Chance of exactly k successes	1	1	—
$n$	Trials Number of independent trials	1	1	1 – 40
$k$	Successes Number of successes counted	1	1	0 – 40
$p$	Success probability Probability of success per trial	1	1	0 – 1

Where it holds

Independent trials with a constant success probability p (Bernoulli trials). Correlated trials or varying p need generalizations (e.g. Poisson-binomial).

Dimensional analysis

P(k) \to

dimensionless probability

C(n,

k)p^k(1-p)^{n-k}

→ number

\times

(number)^k

All factors are pure numbers; the sum over k is 1.

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Discovery

Jacob Bernoulli · 1713

Jacob Bernoulli's posthumous 'Ars Conjectandi' (1713) systematized the distribution of successes in repeated trials and proved the law of large numbers for it. It was the first rigorous treatment of what we now call Bernoulli trials, foundational to all of probability.

Research status: stable

Real-world applications

Quality control and defect-rate estimation
Modeling detector counts and radioactive decay counts
Polling and survey statistics

Common misconceptions

The most probable k is ⌊(n+1)p⌋, not always exactly np
Trials must be independent with fixed p — sampling without replacement gives the hypergeometric instead
It is discrete; the Gaussian is only its large-n approximation

Experimental verification

Coin-flip and dice frequencies, particle-counting statistics, and quality-control defect counts all follow the binomial law within sampling error.

Derivation

Each trial is independent with success probability p.

A specific ordered outcome with k successes and

n-k

failures has probability

p^k(1-p)^{n-k}

.

The number of distinct orderings with k successes is the binomial coefficient C(n,k).

Multiplying gives

P(k)=C(n

,

k)p^k(1-p)^{n-k}

.

Summing over k equals

(p+(1-p))^n = 1

by the binomial theorem — confirming normalization.

Mean np and variance

np(1-p)

follow from linearity over the n trials.

What if…

What if n is large and p small (np fixed)?

The binomial converges to the Poisson distribution with mean $\lambda =np$ — the law of rare events.

What if n is large and p moderate?

It approaches a Gaussian with mean np and variance $np(1-p)$ — the de Moivre–Laplace theorem.

1

Exactly 3 heads in 5 flips

Given ·

n:: 5
p:: 0.5
k:: 3

Find · P(3)

Steps

C(5, $3) = 10$
$p^k(1-p)^{n-k} = 0.5^{5} = 1/32$
$P = 10/32 = 0.3125$

Result · C(5,

3)(0.5)^{3}(0.5)^{2} = 10/32 = 0.3125

Binomial Theorem→Central Limit Theorem→poisson distribution