Will a new gene take over the population?

Question

A population always has exactly N individuals. Each carries one of two versions of a gene, type A or type a, and the two types are equally fit (neither has any advantage). At each step, one individual is chosen at random to reproduce (a copy is made) and one individual is chosen at random to die, keeping the total at N. Suppose there are currently i copies of type A. What is the probability that type A eventually takes over the whole population (reaches N copies) rather than dying out?

Accepted Answer

Let Xt be the number of A-copies. It changes by at most \pm 1 each step, and it stops (is absorbed) at 0 (type A extinct) or N (type A fixed). Because the two types are equally fit, the process has no drift: from state i, the chance the count goes up (A reproduces and an a dies) exactly equals the chance it goes down (a reproduces and an A dies), both equal \frac{i}{N}\cdot\frac{N-i}{N}. When up-moves and down-moves are equally likely, Xt is a martingale: E[X{t+1}\mid Xt] = Xt. Now apply the martingale (fair-game) argument, exactly as in symmetric gambler's ruin. Let hi be the probability of …

Will a new gene take over the population?

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