Calculating Expected Offspring

Problem

For a random variable X taking integer values between 1 and n, the expected value of X is E(X)=∑nk=1k×Pr(X=k). The expected value offers us a way of taking the long-term average of a random variable over a large number of trials.

As a motivating example, let X be the number on a six-sided die. Over a large number of rolls, we should expect to obtain an average of 3.5 on the die (even though it's not possible to roll a 3.5). The formula for expected value confirms that E(X)=∑6k=1k×Pr(X=k)=3.5.

More generally, a random variable for which every one of a number of equally spaced outcomes has the same probability is called a uniform random variable (in the die example, this "equal spacing" is equal to 1). We can generalize our die example to find that if X is a uniform random variable with minimum possible value a and maximum possible value b, then E(X)=a+b2. You may also wish to verify that for the dice example, if Y is the random variable associated with the outcome of a second die roll, then E(X+Y)=7.

Given: Six positive integers, each of which does not exceed 20,000. The integers correspond to the number of couples in a population possessing each genotype pairing for a given factor. In order, the six given integers represent the number of couples having the following genotypes:

AA-AA
AA-Aa
AA-aa
Aa-Aa
Aa-aa
aa-aa
Return: The expected number of offspring displaying the dominant phenotype in the next generation, under the assumption that every couple has exactly two offspring.