In a finite population, allele frequencies fluctuate from generation to generation just by sampling — even with no selection. Eventually one allele wins, the other vanishes. This is genetic drift: evolution by chance, no fitness involved. The Wright-Fisher model — every generation, N offspring sample parents with replacement — predicts how fast fixation happens and what diversity looks like along the way.
why small populations evolve by accident
The Wright-Fisher model (Sewall Wright 1931, R. A. Fisher 1930) is the simplest stochastic model of allele frequencies. Population is constant size N. Generations are non-overlapping. Each individual in generation t+1 picks a parent from generation t uniformly at random (with replacement). With probability p (current allele frequency), they inherit allele A; otherwise allele B.
Mean: 𝔼[p_(t+1) | p_t] = p_t (martingale property — drift is unbiased).
Variance: Var[p_(t+1) | p_t] = p_t(1−p_t) / N.
So drift is faster in small populations. Eventually p hits 0 or 1 — the allele is lost or fixed — and stays there. From starting frequency p₀:
• Probability of fixation = p₀ (without selection).
• Expected time to fixation given fixation ≈ 4N generations (with no selection, large N limit).
Kimura's neutral theory (1968): most molecular evolution is drift, not selection. Synonymous DNA substitutions accumulate at a steady rate proportional to mutation rate alone, independent of population size. The "molecular clock" is the drift signal. This was revolutionary and still controversial.
With selection (coefficient s for the A allele): fixation probability becomes P_fix(p₀) = (1 − e^(−2Nsp₀)) / (1 − e^(−2Ns)) (Kimura 1962). Even slightly beneficial alleles (s > 1/N) get fixed with high probability; slightly deleterious ones (s < −1/N) are usually purged. The product Ns governs which force wins — selection or drift.
Where this rules: founder populations (the Amish, Easter Islanders), bottlenecks (cheetahs, northern elephant seals), bacterial colonies during antibiotic exposure, cancer cell populations, viral quasispecies inside one host. Drift sets the floor on detectable selection — and explains why bad alleles can persist and good ones can vanish.