Charles Darwin is known first and foremost for his notion of natural selection,
the elegant statistical fact that changes in populations
are attributable to the observation that organisms better equipped to handle their environment
are more likely to survive and reproduce, thus passing on their beneficial traits to the
next generation. As a result of natural selection, populations can change greatly over a long time.
A lesser known aspect of Darwin's evolutionary theory dictates how new species
are actually created. Darwin noted that the only way for a population to grow so distinct
that it would actually split off and form a new species would be if
the population were isolated for a very long period.
This notion that isolation forms new species was validated by Darwin's observation that the tiny
Galapagos islands in the South Pacific enjoy a diversity of species rivaling
that of a much larger ecosystem.
Isolated populations also tend to be small, strengthening the effects of genetic drift.
To take an extreme example, consider a population of only 2 organisms that are both heterozygous
for a given factor. Note that there is a 1/8 chance that 2 offspring of these organisms
will possess only recessive alleles or only dominant alleles for the factor,
thus wiping out the other allele completely.
In general, the principle stating that mutations (both positive and negative) can randomly attain
higher proportions in small, isolated communities than they would in large populations,
is known as the founder effect.
An infamous example of the founder effect on human populations occurs in Pennsylvania, where the Amish community
is at risk for a much greater incidence of Ellis-van Creveld syndrome,
a single gene disorder causing a slew of defects, including additional fingers and toes (polydactyly).
The condition has been traced to a single couple in the original Amish settlers, and
it is still preserved in elevated percentages because of the community's isolationism.
In this problem, we would like to apply the Wright-Fisher model of genetic drift
to understand the power of the founder effect. Specifically, we will quantify
the likelihood that an allele will be completely annihilated in a small population after a number of generations.