A population of mice begins with four males and four females. When a heterosexual couple mates, they have an equal chance of producing a male or a female offspring. However, once they produce the first female is born, then the couple becomes barren. What is the gender distribution of mice in seven generations?
Do you think the question intends to ask: what is the gender distribution after the first generation? I feel as if the question does not contain enough information to answer for anything after the first generation. For example, after the first generation, what are the rules if there is not a balanced number of males and females? Are leftover mice just that, leftovers? Or is polygamy allowed? What are the rules of said polygamy?
Considering one couple, $$\sum_{i=1}^{\infty}\frac{1}{2^i}\cdot \frac{2}{i+2}$$ is the expected value of the proportion of females to males. If a couple has $n-1$ males and $1$ female, the probability of such an event is $$\frac{1}{2^n}.$$ The fraction of females to the total number of mice in that family is $$\frac{1 + 1}{n+1 + 1} = \frac{2}{n+2},$$ counting the mother and father. Using Wolfram alpha, $$\sum_{i=1}^{\infty}\frac{1}{2^i}\cdot \frac{2}{i+2} \approx 0.5452.$$ Adding in any number of couples will not change the expected ratio, so 0.5452 is the expected gender distribution of females to males.
this is DJ.
You are correct in attacking the problem by considering each couple. However, there is a bit of inaccuracy in your calculations. I have written the same comments in your assignment which I graded, but I will repeat it here.
First, a caveat: Admittedly, the problem posed is vague regarding the definition of a couple going barren. For example, if Male 1 and Female 1 mate and keep mating until they go barren, can each of them still have children by pairing off with other rats of the opposite sex? For this, let us assume that it is the female mouse that goes barren. With this assumption in mind, the solution should be the same regardless of the rats' social structure, as you will see shortly.
Now, your method for taking the proportion of females in a `family of mice' (a single couple with all its offspring until going barren) has a bit of a problem, which can be best illustrated by example.
Consider the alternate scenario in which, prior to going barren, a couple gives birth to:
--- either 1 male and 1 female with 50% chance
--- or 99997 males and 1 female with 50% chance
If you were to now calculate the expected proportion of females in the population with the same method as you did above, then you would get:
$\frac{1}{2} \cdot \frac{2}{4} + \frac{1}{2} \cdot \frac{2}{100000} \approx \frac{1}{4}$
You may interpret this as, among all possible outcomes for a single family of mice, you'd expect on average for a quarter of the mice in the family to be female.
However, if you were to consider multiple families in a larger population of mice, you would have to address how all of these possible outcomes interact with each other. Among multiple couples of mice, you would expect half of them to give birth to 1 male and 1 female and the other half to give birth to 99997 males and 1 female. The families of mice with 99998 offspring would clearly outweigh the families of mice with only 2 offspring when taking a census of the whole population, and will therefore have a larger influence on the statistic regarding the proportion of females in the total population.
If we are to look at the proportion of females to be expected from a population of mice with a large enough number of couples, we would actually have that:
On average, a family of mice has exactly 2 females.
Also on average, a family of mice has size:
$\frac{1}{2} \cdot 4 + \frac{1}{2} \cdot 100000 = 50,002$
Thus, the expected proportion of females in the population is given by:
$\frac{2}{50,002} \approx 0$.
In other words, you would be hard pressed to randomly select a female from the population the larger it gets.
Now, back to the problem at hand:
We have 4 male-female pairs over 7 generations, and for each generation, a couple's offspring would add to the population's overall male-female ratio. The expected value of the proportion of females in the population can then be expressed as the fraction $\frac{\text{expected number of females in the population}}{\text{expected size of population}}$.
I claim that the fraction we are looking for is $\frac{1}{2}$. Note that we already have $\frac{1}{2}$ of the mice being female to begin with. We would then need to figure out, for a given couple of mice, how many offspring they are to expect (hint: it's going to be 2) and how many female offspring they are to expect (necessarily 1) prior to going barren.