introduction

Markov chain is a mathematical model which depaendsdepend on the probabilities, and is very useful in many science field, especially in genetics. Genetics is the study of genes and is part of biology. Some problems in genetics can be solved using th eMarkov chain. For example if we know the distribution of the present generation, call it $x_0$, we can use the transition matrix $A$ of the population to find genotype distribution $x_n$ of the future genreations.

Let $X_0$ be initial genotype distribution of a population. Assume that the trasition matrix for the genotype of given population is known, call it $A$. One year later, the genotype of the population, $X_1$, is equal the product of the transition matrix A and the inital matrix $X_0$.

$$X_1 = A X_0 .$$ (1)

Two years later, the genotype of the population can be compute similarly,

$$X_2 = A X_1 = A(A X_0 ) = A^2 X_0 .\\$$ (2)

And n years later,

$$X_n = A^n X_0 .\\$$ (3)

According to autosomal inheritance, each inherited trait is assumed to be governed by a set of two genes. We designate these two genes by A and a. Each individual in the population under consideration has two of these genies, as AA, aa or Aa.
Consider eye coloration, in humans. Genotype AA and Aa means having brown eyes and genotype aa means having blue eyes. Gene A in this case is called dominant (or we say A dominates a) and gene a is called recessive (or we say a is recessive to A).