A Problem of Sons

I’m posting this in the Cute Problems folder, but I’m mainly putting it up here as a sort of experiment. This little puzzle was posted on Twitter by someone I follow and it got a huge number of responses (>25,000). I was fascinated by the replies, and I’m really interested to see whether the distribution of responses from readers of this blog is different.

Anyway, here it is, exactly as posted on Twitter:

Assume there is a 50:50 chance of any child being male or female.

Now assume four generations, all other things being equal.

What are the odds of a son being a son of a son of a son?

Please choose an answer from those below:

 

UPDATE: The answer is below:

 

The correct answer is D. Ceteris paribus, every son has a father who is a son of a father who is a son, etc. The first two statements in the puzzle are irrelevant, and are there merely to make you think the question has something to do with probability. It doesn’t. It’s a question of comprehension. For the record, the person who posted this question is a not a mathematician or a statistician but a lawyer. Only 51% of his respondents got the correct answer, whereas over 70% of mine have (so far).

I think the only part of the puzzle you might quibble about is the `all else being equal’, which means that it is possible to think of exceptions, e.g. in this case same-sex parents, but these should not be taken to negate the generality.

 

 

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25 Responses to “A Problem of Sons”

  1. It is a graph tree where the left edge is the son, and the right edge is the daughter: it exist only a set of only left edge (descending along the family tree), and there are 2^4=16 children of the same generation. I think that the probability is 1/16.

  2. I voted. To some extent it is intentionally formulated to provoke wrong answers, but it is also not well defined in an important sense.

    • telescoper Says:

      What sense is that? It seems completely well-defined to me.

      • I’ll reply when you reveal the answers.

      • I need a Keplerian cryptic to prove that I voted correctly before the result was known. Of course, Kepler wrote in Latin, which is not popular anymore in these parts. I’m sure that all sane people, if they think correctly, will conclude that, after eliminating all the wrong answers, the only one left must be correct.

  3. “What are the odds of a son being a son…”

  4. I vote for RON (re-open nominations).

  5. Simon Kemp Says:

    It doesn’t specify the number of children per generation, implying 100%. But if there are 4 generations, the first one doesn’t count as being a ‘son’ of someone, implying 0%.

  6. James Dunlop Says:

    Poorly defined question – not clear from the wording where the generations start. Certainly wouldn’t get through any sensible exam panel.

  7. James Dunlop Says:

    Problem is not cute or clever in my opinion. Answer is 1/16 or 1/8 depending on how you read the question, which is very badly written.

    • telescoper Says:

      It’s neither of those answers.

      • James Dunlop Says:

        Hmm – still think it is poorly written – I read it as starting with a son (e.g. my great-grandfather) and connecting the son of son chain to my son through 4 generations. Still think it can be read that way. Wouldn’t make it through our exam checking meeting for sure.

      • James Dunlop Says:

        Of course lawyers are experts at writing things in ways that are open to interpretation. That’s what keeps them in a job. I would still argue there is no single right answer to this question, as it relies on how you view the word “a”. Sometimes “a” is the same as “any”, but sometimes it means “a particular” – which is the interpretation that the preamble is designed to point you towards. But even without the preamble, that is a fair interpretation – e.g. if I say you need to get a bus from my house to the railway station, I don’t mean any bus. Language is more nuanced than computer code.

      • Phillip Helbig Says:

        I agree that it is intentionally poorly written, but “that a son is a son” means that this fact is given, so the question is only about 2 generations, not 4, hence 25% is the correct answer.

  8. “I’ll wait until I get a decent number of responses before revealing the correct answer.”

    What is a decent number?

  9. Will Grainger Says:

    I look forward to seeing the correct answer when you are back from St Ives.

  10. Lucas Fowler Says:

    “Only 51% of his respondents got the correct answer, whereas over 70% of mine have (so far).”

    But you posted this after David Allen Green’s poll had had over 37’000 votes and he also has over 140’000 followers, many of whom will have read the resulting thread without voting. So given that you can assume that you both have a number of twitter followers in common I’m not sure how indicative your poll is.

    • telescoper Says:

      Does that matter?

      • Lucas Fowler Says:

        Well yes, because “It’s a question of comprehension.”. Many people fixate on the sentences implying a probability calculation, but seeing any of the discussion will promote reflection leading to a different answer.

        As I was going to St. Ives … is in a similar vein.

  11. Anton Garrett Says:

    The first two statements in the puzzle are irrelevant, and are there merely to make you think the question has something to do with probability. It doesn’t. It’s a question of comprehension. For the record, the person who posted this question is a not a mathematician or a statistician but a lawyer.

    But virtually all of the pioneers of the quantification of probability theory, several hundred years ago, were lawyers or sons of lawyers, obviously because law is to do with the probability of guilt or innocence given the evidence.

  12. Phillip Helbig Says:

    “every son has a father who is a son of a father who is a son, etc”

    Yes, but the wording is ambiguous. “What are the odds of a son being a son”: this much is irrelevant, since it assumes that we are talking about a son. “of a son of a son?” is the interesting part. At least under normal assumptions, every son is the son of a son and so on, so this is trivial. When talking about “generations”, one normally is talking about children, grandchildren, etc, but in generation X+1 not the other parent from generation X. So, for someone down the line from the original couple in the first generation, the question is whether he is the son of a son of this older generation. The probability here is clearly 1/4, since the probability is 1/2 that his father is a descendant of the original generation and 1/2 that is mother is (neglecting the possibility of incest, of course). The same argument goes for his paternal grandfather.

    Hence, Your Honour, the proper answer is 25%. I rest my case.

  13. Simon Kemp Says:

    I still say if there are 4 generations there can only be 3 sons. I’m sure a lawyer could defend that statement.

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