Geometric distribution

Flip a biased coin with success probability p until the first head. The number of flips needed follows a Geometric distribution. One parameter does all the work.

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

p (success prob) 0.30 n (samples) 500

Dark bars: PMF

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

. Light bars: empirical frequencies from 500 samples.

E [X] = 1/ p

= 3.3 trials.

What to notice

Small p produces a heavy right tail — most runs take many trials, but occasionally the first flip succeeds.
Large p pushes almost all probability onto k=1 and k=2 — success comes quickly.
Mean $E [X] = 1/ p$ and variance $Var (X) = (1 - p) / p^{2}$ both grow as p shrinks.

Memorylessness

The geometric distribution has a striking property: past failures tell you nothing about future success.

P (X > m + n ∣ X > m) = P (X > n)

If you’ve already failed m times, the distribution of remaining trials looks exactly like starting fresh. This is the only discrete distribution with this property — just as the exponential is the only continuous memoryless distribution.

Connection to the exponential

In a Poisson process with rate λ, the number of trials in a time interval of length 1/λ follows (approximately) a geometric distribution with p = λ/λ = 1. More precisely: if you discretize a Poisson process into steps of size δ, each step succeeds with probability p = λδ, and as δ → 0 the geometric distribution converges to the exponential. They are the same memorylessness expressed in discrete vs continuous time.