Almost absurd literary math puzzle
What is the probability that someone is an only son of an only son of an only son?
What's the probability that this person is himself an only son? I don't know that it's logically derivable - couples have a variety of numbers of children, of various mixes of genders. So far as I can see, that probability can be measured but not derived. Call that probability P.
If I had to estimate it, I'd say that the most common number of children is two, and of the four possibilities (BB, BG, GB, and GG), half of the boys are only sons. (This is presuming that a sister doesn't ruin "only son" status.) So that leaves the probability at 50%. Tweak that with only children (100% of male only children are the only son) vs. more than 2 children (reduces the probability of being an only son), but the probability is reasonably somewhere in the neighborhood of 50%.
The probability we are looking for is P^3 (presuming that only-son-ness is not something hereditable, that is, that the probabilities are independent in each generation. So the total probability is something like 1/8 = 0.125 (or 1/16 if we don't know that the "someone" is male).
Have I missed something here?
Another line of thought, possibly just a line of confusion:
Assume "someone", S, is the OS of an OS of an OS. Look at the siblings of S's grandfather - he could have had any number of sisters, but had no brothers. The probability of that situation occurring goes down with each increase in the number of sisters. So maybe an exact answer can be given, a formula including terms for the number of sisters each of the 3 OSs have? (not forgetting the probability that S is also female)
Yes, that's what my point was - "he could have had any number of sisters" - if you have 12 sisters, it's extremely unlikely you are an only son. The more sisters, the more unlikely being an only son is.