# Coalescent Theory - Probability for $k$ alleles that of one coalescence event occured $t+1$ generations ago

From this textbook

Under the wright-Fisher model of genetic drift and under the assumption that all alleles are neutral, the probability that $$k$$ alleles had $$k$$ distinct parent alleles the previous generation is

$$Pr(k) = \prod_{i=1}^{k-1}1-\frac{i}{2N} ≈ 1-\frac{{k \choose 2}}{2N}$$

The chance of two alleles not coalescing for $$t$$ generations is $$\left(1-\frac{1}{2N}\right)^t$$ , and the chance that they coalesce in the next generation is $$\frac{1}{2N}$$. Therefore, the probability that 2 alleles had a common ancestor t+1 generations ago is

$$\frac{1}{2N}\left( 1-\frac{1}{2N} \right)^t ≈ \frac{1}{2N}e^{\frac{-t}{2N}}$$

I understand up to this point!

The probability that the $$k$$ alleles do not coalesce for $$t$$ generations, and then one pair coalesce to give $$k-1$$ alleles at $$t+1$$ generations ago is as follows:

$$Pr(k)^t \left[ 1-Pr(k) \right] ≈ \frac{{k \choose 2}}{2N}exp\left[ -\frac{{k \choose 2}}{2N}t \right]$$

Can you help me to understand this last part? (both the left and right part of the equation)

Is this the exact text from the book? The left side seems to represent the probability for

"No coalescence in $k$ lines in $t$ generations (i.e. the $Pr(k)^t$ term), and at least one coalescence among those lines in generation $t+1$ (the $1-Pr(k)$ term)"

which is the same event as

"First coalescence event in $k$ lines is exactly in generation $t+1$".

The right hand side is derived analogous to the second equation, with $1-\frac{1}{2N}$ ("no coalescence in two lines") replaced by $1-\frac{k\choose2}{2N}$ (approximation for "no coalescence in $k$ lines):

\begin{align} Pr(k)^t \left[ 1-Pr(k) \right] &≈ \left( 1-\frac{k\choose2}{2N} \right)^t \left [1 - 1 + \frac{k\choose2}{2N} \right] \\ &\approx \exp\left( - \frac{k\choose2}{2N} t \right) \frac{k\choose2}{2N} \end{align}

were is first approximation is due to your first calculation. The second one seems to use a first order Tailor approximation of $\exp(x)$:

$$\exp(-xt) = exp(-x)^t \approx (1-x)^t$$

• +1 Ok, I got the idea. No, that is not exactly the same text than in the book. I summarized it. Thanks a lot for you answer. The answer would be perfect if you could just show that $$\frac{1}{2N}\left( \prod_{i=1}^{k-1}1-\frac{i}{2N} \right)^t ≈ \frac{{k \choose 2}}{2N}exp\left[ -\frac{{k \choose 2}}{2N}t \right]$$ – Remi.b Apr 15 '14 at 14:37
• I extended my answer to show the equation. I have never seen this approximation of $\exp$ before... – Paul Staab Apr 15 '14 at 15:13