It seems to me that they are all Google quality.
The final quantity sought in the question is the proportion of boys to girls. Given that:
- The probability of a child of a given sex is 0.5 for any given pregnancy;
- The probability of a child of a given sex is independent of the outcome of any other pregnancy;
...then it comes down to the same thing as people tossing a coin and stopping when it lands on heads. If x people toss a coin, then x/2 will land on heads and stop tossing, and x/2 will land on tails and continue to toss. (Ooh-er, Missis!)
So the next cycle will be (x/2)/2 land on heads, and (x/2)/2 get tails.
And then ((x/2)/2)/2 get heads and ((x/2)/2)/2 get tails.
The probabilities of each outcome remain equal whatever number of iterations you choose, as no previous iteration can affect the outcome of a subsequent iteration, and no single outcome within a given iteration can affect another outcome within the same iteration. Although the value of x is halved on each iteration (and is therefore the only variable quantity), the probability of each outcome remains at 0.5 for each iteration. As the only variable is independent of the relative proportions of the outcome, it is obvious that the relative proportions will remain unaltered as the series is summed: 0.5 heads, 0.5 tails.
I don't know if Google actually use this question (I doubt it - there's a lot of crap spread around about interview questions used by top companies) but insofar as it has any interest, it's in that by introducing the question of human reproduction it might throw people off the scent, as people have a vague feeling that it must be quite complicated if it involves something so important. This is a purely psychological thing, and given that the question can be equally well put in terms of coin-tossing, it is I suppose possible that it might be used to see if a victim can avoid being distracted by irrelevant circumstances and focus on the underlying principles.
But to be honest, if I went for an interview at somewhere like Google and they started asking me crap like this, I'd tell them to stop pissing about and get on with it.
