# Equation relating probability of fixation and number of generations

I am trying to find an equation that relates the variables of probability of fixation and generations. Or, how does number of generations affect the probability a gene will fix, if population size doesn't significantly decrease or increase.

• You may want to have a look at the notion of Moran process, that's pretty much what these models are designed for. – Mowgli Dec 6 '18 at 4:21

From Hartl and Clark's 'Principles of Population Genetics' Third Edition chapter 7: Kimura and Ohta (1969) showed that the mean time in generations until the allele of frequency p is fixed (ignoring cases where the allele is lost) is:

$$t_1(p) = -4N[(1-p)log(1-p)]/p$$

The mean time to loss of the allele is:

$$t_0(p) = -4N[p \cdot log(p)]/(1-p)$$

Thus the mean persistence time of an allele is:

$$t(p) = -4N[p \cdot log(p)+(1-p)log(1-p)]$$

Hopefully I didn't make any typos there, but I reccommend this book to learn all the ins and outs of popgen. There is a PDF of the third edition online in an archive. For more book recommendations, please have a look at the post Books on population or evolutionary genetics?

You can also find these equations in the original article: Kimura and Otha (1968)

• By wrapping your equations in dollar signs, you can use LaTex type of formatting. For example $\frac{1}{\sqrt{2}}$ becomes $\frac{1}{\sqrt{2}}$. See your answer edited for more info. I also added the original article and a link to more book recommendations. – Remi.b Dec 6 '18 at 21:14

It seems like answer to your question can be related to effective population size and idealized population. One of the models to to consider is Wright-Fisher model: $$\frac{(2N)!}{k!(2N-k)!)}p^{k}q(2N-k))$$ where N is individuals, so there are 2N copies of each gene, k is number of copies of an allele that had frequency p.