A simple model for the spread of mutant genes goes like this.

We have a population of N animals that reproduce asexually. In every generation, they produce a collective total of N offspring between them, then die themselves. This process continues, so every generation consists of a fixed total of N animals.

This model is similar to the famous Wright–Fisher model, which is the foundational model of the science of population genetics. These models often aren’t very realistic, but they give us useful insights that are reflected in much more complicated scenarios in real life.

A genetic mutation arises, so every animal has one of two genetic alleles for a particular trait: a safe allele A, or a mutant allele B. To start off with, just one of the N animals has the mutant allele B.

Suppose there are b animals with allele B in some generation.
The genetic inheritance model assumes that Number of B alleles in next generation ∼ Binomial 
N, b N 
.

a. Suppose there are N = 20 animals in the population, and just one of them has the mutant allele B. What is the probability the mutant allele has died out by the next generation?

b. If either allele dies out in any generation, can it be revived in a future generation? Explain your answer.

c. Suppose 10 of the 20 animals currently have the mutant allele B. What is the distribution of the number of B alleles in the next generation? Find the probability that the next generation has at least 12 animals with the mutant allele.

d. The process by which the mutant allele proportion changes from one generation to the next is called genetic drift. It is an important consideration for small populations, because harmful mutant genes can quite easily drift to the point where the safe allele is extinct and the mutant allele infects the whole population. In fact, this happens with exactly This is called genetic fixation.
probability 1/N, if the process starts with just one mutant.
With probability (N − 1)/N, it will become fixed for the safe allele instead.
If N = 20, write down the probability that the population ends up fixed for the mutant allele B, starting from just one mutant.

e. Now suppose we’re looking not just at one genetic trait, but at 500 genetic traits, still with a population of size N = 20. The drift processes for different genetic traits can be considered independent. Suppose the population continues reproducing indefinitely, and at some point a single mutant arises for each of these 500 traits. Let X be the number of these genetic traits that ends up fixed for the mutant allele. What is the distribution of X, and what is the peak value of this distribution? What does this mean in practice?
The potential is high for things to go wrong in small populations. There’s even a practice called "genetic rescue" to mitigate this risk for highly endangered species.