Mendel's First Law

This time we are asked to calculate the probability that the offspring of two people in a given population receives a dominant allele for a trait. The people in the population are either homozygous dominant (k), heterozygous (m), or homozygous recessive (n) for the trait.

To solve this problem I felt that the easiest way would be to derive an equation for the probability that the offspring gets a dominant allele and then make a program that makes the calculation based on this equation.

The equation must take two things in consideration. First, it needs to include the probability that each parents can have a set of alleles that is either k, m or n (that is AA, Aa or aa, where A is dominant and a is recessive). Then it also needs to consider the probability of each possible couple receiving a child with one or two dominant alleles. For example, if either of the parents is homozygous dominant, then the probability that the child will have at least one dominant allele is 1, but if both parents are homozygous recessive the probability is 0.

To set up the equation I started drawing the possible combinations of parents and the probability for each of them. Below is a sketch of this (pop = total population = k + m + n).

To receive the probability of a specific couple being randomly selected you simply multiply the probability for the first selection event with that of the second selection event. For exmple, the probability that both parents are homozygous dominant is (k/pop)((k-1)/(pop-1)).

After this step I also needed to consider the probability that the offspring actually gets the dominant allele. This is quite easily figured out using a Punnett square. Then you just multiply the probability for the couple with the probability of them being randomly selected. To get the overall probability of the offspring of a randomly selected couple having a dominant allele you just add them all together.

After simplifying the equation, this is what I ended up with:

Then it was just a matter of writing a simple program to make the calculation given k, m and n. The following is what I ended up with, including the values of k, m and n that I received from Rosalind.

k = 23                                                        

m = 26                                                        

n = 22                                                        

pop = k + m + n                                               

prob = (4*(k*(k-1)+2*k*m+2*k*n+m*n)+3*m*(m-1))/(4*pop*(pop-1))

print(prob)