Editing Genetic drift (section)

==Probability and allele frequency==
The mechanisms of genetic drift can be illustrated with a very simple example. Consider a very large colony of [[bacteria]] isolated in a drop of solution. The bacteria are genetically identical except for a single gene with two alleles labeled '''A''' and '''B''', which are neutral alleles, meaning that they do not affect the bacteria's ability to survive and reproduce; all bacteria in this colony are equally likely to survive and reproduce. Suppose that half the bacteria have allele '''A''' and the other half have allele '''B'''. Thus, '''A''' and '''B''' each has an allele frequency of 1/2.

The drop of solution then shrinks until it has only enough food to sustain four bacteria. All other bacteria die without reproducing. Among the four that survive, 16 possible [[combination]]s for the '''A''' and '''B''' alleles exist:  <br /> (A-A-A-A), (B-A-A-A), (A-B-A-A), (B-B-A-A), <br /> (A-A-B-A), (B-A-B-A), (A-B-B-A), (B-B-B-A),<br /> (A-A-A-B), (B-A-A-B), (A-B-A-B), (B-B-A-B),<br /> (A-A-B-B), (B-A-B-B), (A-B-B-B), (B-B-B-B).

Since all bacteria in the original solution are equally likely to survive when the solution shrinks, the four survivors are a random sample from the original colony. The [[probability]] that each of the four survivors has a given allele is 1/2, and so the probability that any particular allele combination occurs when the solution shrinks is

:<math>
\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{16}.
</math>

(The original population size is so large that the sampling effectively happens with replacement). In other words, each of the 16 possible allele combinations is equally likely to occur, with probability 1/16.

Counting the combinations with the same number of '''A''' and '''B''' gives the following table:
{| class=wikitable
|-
| '''A'''
| '''B'''
| Combinations
| Probability
|-
| 4
| 0
| 1
| 1/16
|-
| 3
| 1
| 4
| 4/16
|-
| 2
| 2
| 6
| 6/16
|-
| 1
| 3
| 4
| 4/16
|-
| 0
| 4
| 1
| 1/16
|}

As shown in the table, the total number of combinations that have the same number of '''A''' alleles as of '''B''' alleles is six, and the probability of this combination is 6/16. The total number of other combinations is ten, so the probability of unequal number of '''A''' and '''B''' alleles is 10/16. Thus, although the original colony began with an equal number of '''A''' and '''B''' alleles, quite possibly, the number of alleles in the remaining population of four members will not be equal. The situation of equal numbers is actually less likely than unequal numbers. In the latter case, genetic drift has occurred because the population's allele frequencies have changed due to random sampling. In this example, the population contracted to just four random survivors, a phenomenon known as a [[population bottleneck]].

The probabilities for the number of copies of allele '''A''' (or '''B''') that survive (given in the last column of the above table) can be calculated directly from the [[binomial distribution]], where the "success" probability (probability of a given allele being present) is 1/2 (i.e., the probability that there are ''k'' copies of '''A''' (or '''B''') alleles in the combination) is given by:

:<math>
{n\choose k} \left(\frac{1}{2}\right)^k \left(1-\frac{1}{2}\right)^{n-k}={n\choose k} \left(\frac{1}{2}\right)^n\!
</math>

where ''n=4'' is the number of surviving bacteria.