Chance in population genetics

Random changes in allele frequencies between generations are called "genetic drift".  Although genetic drift is always at work and interacts with natural selection processes, it is best understood by considering it alone. Hence, we shall assume that all the alleles have neutral fitness. 

Suppose we have a population with two alleles, a and b.  In every generation, a has a small chance of increasing its frequency with respect to b.  If a's luck continues long enough, a will become fixed, i.e., it will be the only allele.  If this happens, then the population becomes homozygous and, of course, once homozygous it will remain so.  Once genetic drift is allowed, the probabilities are not given by Hardy-Weinberg.  For example, suppose that initially there are 5 a's and 5 b's.  However, by pure chance, in the next generation relative proportions have shifted to 6 a's and 4 b's.  Note that the chances of getting more a’s than b's in the third generation has gone up.  Of course, now b may become lucky and the process may reverse direction.  However, it is only a matter of time before either a or b becomes very lucky and (other things being equal) ends up by being the only allele.

The same idea applies to mutations.  Suppose that a mutation appears with frequency f, say 5%.  Then, if there are N individuals, there will be 2N individual alleles, two per locus.  Some of these will be transmitted to the next generation and some won't.  For example, most of the individuals with a certain allele a might be accidentally killed, thus lowering its probability of transmission.  Or an individual allele might just be unlucky and appear in the next generation with a frequency lower than Hardy-Weinberg's.  The crucial point here is that once an individual allele (call it "John") is not transmitted, it vanishes forever.  Hence, every generation some individual allele or other may just not be transmitted and therefore drop out, as it were.  Hence, given enough time, all the descendent allele will come from a single allele (a single John, as it were), called "the coalescent".  As a result, pure genetic drift results in homozygosity.  Note that since only genetic drift is allowed (only pure chance), each of the original individual alleles has the same chance to become the coalescent, which means that the probability that an individual allele (our original John) has a probability of 1/2N of being the coalescent and a probability of (2N-1)/2N of vanishing. 

Suppose now that the mutation rate for an allele a is f; that is, suppose that of the original 2N individual alleles, 2Nf are mutant.  For example, if f=5% and the population is made up of 100 individuals (N=100), then 10 alleles will be mutant.  The probability that one of these mutants will be come the coalescent is (2Nf)/2N=f , which means, amazingly, that the chance that a mutation becomes fixed depends on its frequency alone, and not on the size of the population, as one would think, given that chance fluctuations are more powerful in small  populations.