Peter Winkler shares some great variations of this: "Boy Born on Tuesday" (p. xix) and "Men with Sisters" (p. xxii) in "Mathematical Puzzles".

"Mrs. Chance has two children of different ages. At least one of them is a boy born on Tuesday. What is the probability that both of them are boys?"

(note: it is a puzzle, not a biology or data demography problem. so there are 50/50 independence assumptions on gender and uniform day of week assumptions prior to adding the conditioning.)

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layer8 4 days ago | parent | next [-]

Here “on Tuesday” is ambiguous, in my opinion. I first thought it meant “on a Tuesday” and that it was just a diversion. But it is likely intended to mean “last Tuesday” or “this Tuesday” (which excludes the boy-then-girl case). Wording it more clearly would likely reduce the ratio of wrong answers.

Furthermore, “of different ages” is likely intended to exclude the case of twins. However, even with twins, one is generally nominally older than the other. (Not to mention that it’s possible for two non-twin siblings to be the same age in years, at certain points in time.) Why not just say “that aren’t twins”?

I loathe when logic puzzles are obscured by ambiguous language, turning them more into “gotcha” text interpretation riddles than logic puzzles.

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jmount 4 days ago | parent | next [-]

Puzzles are definitely odd birds. I myself have gotten into a literal screaming match try to push my belief that they never should be used in interviews. The bulk of that was an interviewer said the interviewee was "clearly confused when they were asked a puzzle" yet refused to agree that may evidence the presentation of the puzzle may in fact be confusing (and not measuring anything).

I can't speak for Winkler, but both he and Jaynes implicitly separate the reading of the puzzle from the work. Winkler start his book with a few awful "reading trick ones", but in the explanations gives a few reading directions to try and avoid that going forward. I happen to know he meant "on a Tuesday." But a correct solution to a different read would be a correct solution even if it doesn't match the book text. I don't think he was trying to set a text trap, it is just hard to be clear, concise, and unambiguous at the same time. (Even "on a Tuesday" isn't completely clear if it means "all I am telling you was the day of week was Tuesday" versus "it was a very specific Tuesday, that I am not telling.")

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exmadscientist 4 days ago | parent [-]

The value of puzzles in interviewing is never about reaching the solution. It is about seeing how candidates deal with tricky situations that stretch them a bit, because that happens all the time on the job. It should almost always be done interactively, so you can see what clarifications and extra information they ask for, when and how they give up, and if they're dumb enough to say HR violations out loud (it does happen).

This does require a rather skilled interviewer, so the benefits may well not be worth it. But it can be very interesting information to have.

	▲	teekert 4 days ago \| parent [-]
		Or whether you watch Veritasium and just know you can jump out of that blender because weight scales with the cube of height and muscle power scales quadratically.

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zeroonetwothree 4 days ago | parent | prev | next [-]

I think it clearly means “on a Tuesday”, anything else wouldn’t make sense as a puzzle. We are meant to assume each day of the week is equally likely.

Excluding twins is so that we can assume the probability of each day of the week is independent.

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stronglikedan 4 days ago | parent | prev [-]

I agree with the first one, but age is measured in days at a minimum, so twins are always the same age. (I'm sure there are cases where they are born farther apart due to some issue with the pregnancy, but that is statistically insignificant here.)

	▲	layer8 4 days ago \| parent [-]
		Even if you take days, one can be born at 23:58 and the other at 00:03 the next day. (And it could be New Year’s day — in some cultures that would even imply different ages in years.) Regardless of days, it’s not uncommon to talk about who is the older twin. Of course colloquially twins are the same age, but we are talking about a mathematical puzzle about probabilities here, where precision is paramount.

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jimmaswell 4 days ago | parent | prev | next [-]

Why can't you just disregard the existing boy and reframe the question as the probability that the other child is a boy, and the space of all possible answers is BG and BB, equally probable (1/2)? Not really following explanations I find online.

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joshuaissac 4 days ago | parent [-]

Because GB is also a possibility. You are not told that the existing boy is the elder child.

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jimmaswell 4 days ago | parent [-]

https://www.online-python.com/Q5leTWuvb6

https://www.online-python.com/RueVd2514m

No matter how I frame or interpret this question, the birthdays and birth order appear completely irrelevant - the results are still ~0.50 as expected. Whatever the author was trying to say, they didn't communicate it well. I'm really curious exactly what word or phrase the author thought I was supposed to take to mean something else. If someone could edit one of these simulations to show what the author intended then that would probably be the clearest way.

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LudwigNagasena 4 days ago | parent [-]

https://www.online-python.com/6LDtZz7lMh

* Take all families with two children

* Take a subset where at least one child is a boy born on Tuesday

* Take a subset of the previous subset where all children are boys

* The share of the 2nd subset relative to the 1st subset is around 48%

	▲	jimmaswell a day ago \| parent [-]
		Yeah, makes sense. I thought I saw someone say 1/3 which is what seemed impossible.

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zeroonetwothree 4 days ago | parent | prev | next [-]

So we would write down the possibilities as (B-Tue, B-Tue), (B-Tue, G-Any), (B-Tue, B-NotTue) and the inverse for the latter two. This results in 27 cases. Of those 13 have two boys so the answer is 13/27.

	▲	zeroonetwothree 4 days ago \| parent [-]
		Similarly if you had the knowledge “at least one is a boy born on May 11” then it would be very close to but slightly less than 50%. So we can see in the limit as the information becomes more and more specific it turns into the unconditional probability. That is, the case of “the first is a boy, what is the probability both are boys” (50%). I think this clarifies the situation in the OP pretty well.

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Rickasaurus 4 days ago | parent | prev [-]

Great book, I highly recommend it too.