A population problem

Really a variation on a well-known theme, inspired by this and also recently appearing here.

Suppose there is an initially finite population consisting of males and females which breeds without multiple births according to the following rules. Each femaile continues to produce offspring until a male child is borne, and then ceases to breed. Show that (even in this ideal universe) with probability 1, a point in time will be reached after which no further reproduction is possible. We assume that births are iid with equal probability of a male or a female being born.

I suggest this as an exercise, and include a solution below the fold.

***SPOILER ALERT***

WLOG there are initially zero males. WLOG noone ever dies. Then at all times, there is an injective function from the set of males to the set of females, given by taking ones mother. Consider the difference F-M between the number of females and the number of males. Over time, this difference evolves as a random walk on the integers, at each stage adding one with probability 1/2 and subtracting one with probability 1/2. This random walk has the property that it will visit every integer with probability 1. (It is folklore knowledge that the corresponding result is true in two dimensions and false in three). Hence this value F-M reaches a non-positive number with probability 1, at which point no more births are possible, since every female already has a male son.

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