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10,000

= 0.0004. Using the exponential approximation instead, 1 - e ia°°° = 0.00039992 as the chance that a pair of lineages experiences a coalescence at or before four generations elapses. Thus, the exact probability and the approximation are in close agreement. The exponential approximation of the exact probability improves as the population size increases. See Hein et al. (2005) and Wakeley (2008) for a more detailed discussion of the relationship between population size and the error of the exponential approximation of coalescence probabilities.

Approximating probabilities of coalescence with the exponential distribution makes computing more practical and also yields several generalizations about the coalescence process. In particular, the geometric and exponential probability distributions can be used to obtain an approximate average and variance for coalescence times. It turns out that for both types of distribution the mean time to an event is simply the inverse of the probability of an event occurring. In the coalescence process, the probability of coalescence 1

for a pair of lineages is so the average time that elapses until coalescence is 2N when the coalescence process is approximated with the exponential distribution. The average time to a coalescent event is often called the waiting time. Another generalization is that the range of individual coalescence times around that average is quite large. Based on the exponential distribution, the variance in the waiting time is 4N2 so that range of coalescence times around the mean grows rapidly as the size of the population increases. Thus, the length of branches connecting lineages to their ancestors will be highly variable about their mean value. This can be seen in Fig. 3.25, which shows six independent realizations of the coalescent tree for six lineages.

It is possible to determine the average time for more than two lineages to find their MRCA. Suppose we want to determine the waiting time for k lineages,

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