Bernoulli distribution

A single trial with two outcomes: 1 with probability $p$ , 0 otherwise. Every other discrete distribution on this site — Binomial, Geometric, Negative Binomial — is built out of independent Bernoullis.

P (k; p) = p^{k} (1 - p)^{1 - k}, k \in {0, 1}

p (success prob) 0.30 n (draws) 200

Dark bars: PMF

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

. Light bars: empirical frequencies from 200 draws. Mean

p

, variance

p (1 - p)

What to notice

Only one knob. The whole distribution is determined by $p$ . The dark bars show the exact PMF; the light bars show the empirical frequencies. Raise n and the two converge — that’s the Law of Large Numbers.
Variance is maximized at p = 0.5, where you’re least sure of the outcome, and shrinks to zero at p = 0 or p = 1.
Summing n independent Bernoullis gives a Binomial(n, p). That’s the bridge from this one-bit coin to everything else discrete.

Why it matters

The Bernoulli is the indicator function for any yes/no event. Expressing $E [X] = p$ as “the probability equals the expected value of the indicator” is the move that underlies most elementary probability identities.