# Probability of Extinction under Genetic Drift

Here is the Wright-Fisher model of genetic drift:

$$\frac{(2N)!}{k!(2N-k)!}p^kq^{2N-k} \Leftrightarrow \binom{2N}{k}p^kq^{2N-k}$$

where $\binom{2N}{k}$ is the binomial coefficient.

This formula gives the probability of obtaining $k$ copies of an allele at generation $t+1$ given that there are $p$ copies of this allele at generation $t$. $N$ is the population size and $2N$ is the number of copies of each gene (this model applies to diploid population only).

From this formula, how can we calculate the probability of extinction of an allele in say 120 generations starting at a given frequency, let's say 0.2?

and

How can we calculate the probability of extinction rather than fixation of an allele present at frequency $p$ if we wait an infinite amount of time?

– Did
Dec 5 '13 at 6:54

update

Kimura and Ohta (1969) showed that assuming an initial frequency of $p$, the mean time to fixation $\bar t_1(p)$ is:

$$\bar t_1(p)=-4N\left(\frac{1-p}{p}\right)ln(1-p)$$

similarly they showed that the mean time to loss $\bar t_0(p)$ is

$$\bar t_0(p)=-4N\left(\frac{p}{1-p}\right)ln(p)$$

Combining the two, they found that the mean persistence time of an allele $\bar t(p)$ is given by $\bar t(p) = (1-p)\bar t_0(p) + p\bar t_1(p)$ which equals

$$\bar t(p)=-4N\cdot \left((1-p)\cdot ln(1-p)+p\cdot ln(p)\right)$$

This does not answer any of the two questions!

• the average persistence time

but not...

• the probability of fixation rather than extinction if we wait an infinite amount of time

neither...

• the probability that the allele get extinct over a period of say 120 generations.

• The probability of fixation given infinite time is just $p$ (the frequency of the allele at that time) - I just ran a simulation in R with initial frequency of 0.2 (1000 replicates over 5000 generations $N$ =10000 and $p$ =0.2) and the average fixation rate was 0.201 (though simulations with smaller populations are more sensitive to stochasticity thus vary more around $p$ but are still very close, $N$ =10 had a fixation of 0.184) ... I'm sure there must be a way to calculate the probability after a given number of generations... Nov 29 '13 at 10:48