# A bizarre question asked at a job interview about conditional probability [closed]

I was recently asked a "logic puzzle" type question at a job interview which seems to be a slightly mangled version of a N-coin puzzle (intended to test the subject's knowledge of conditional probability).

You are approached at a party by a shrouded figure who informs you that he/she has a brother. Next he/she asks you to guess his gender. What is the probability that he/she is actually male?

It occurred to me that the interviwer had got the question wrong - this seemed like a mangled variant of a more common math-test question where N hiden coins are concealed, n of which are heads-up. If I reveal only one of the n coins and it turns out to be heads-up, what's the probability that the next coin I reveal is also showig heads.

But it also occurred to me that I might have misunderstood some kind of subtlety in the question:

I claimed in the interview that in the original puzzle, knowing the gender of the mystery-person's sibling provides no clues at all as to the mystery person's gender since gender is an independantly determined value, however is this some kind of puzzle with a counter-intuative answer (like the Monty-Hall problem)?

• While definitely interesting, this would seem to be about logic for logic's sake. As per meta discussions, this is probably more appropriate for math.se Feb 7, 2012 at 17:32
• I was not sure where to post it really - it's a question about a puzzle which may or may not be broken. Yes, it pertains to logic but also lack of logic and confusion. Math.se seems to be non-English - so perhaps there's something like a stacksite for this kind of question. (reddit math?). Feb 7, 2012 at 18:04
• Math.SE claims it is for mathematics at any level; if it's alright with you, I can talk to the mods over there about migrating this that way Feb 7, 2012 at 18:46
• Indeed, this sort of question gets asked in various reincarnations every now and then on Math.SE (which I also read regularly). But I went ahead and wrote up an answer here as well. Feb 7, 2012 at 21:44
• As formulated, the question you were asked is a biology question, not math nor philosophy. Feb 8, 2012 at 18:14

In my opinion, you were asked a very good question about when NOT to use conditional probability.

The first thing to think about when being tempted to use conditional probability is: are the two events actually correlated?

If they are correlated, you can go on and do all the complicated math.

If they are not correlated, that is, if the two events are totally independant, then you only have to care about the probability of the one event, regardless of the other.

In the problem you were given, the fact that the shrouded person has a brother is totally independant from his/her sex determination. These are totally not correlated. So the probability that he/she is male/female is the same as for the rest of the human population: about 50,3% for male 49,7% for female.

• You are very much welcome. it took me 30 minutes to put it together in a consistent way, I imagine you did not have 30 minutes to answer during your interview ;) Feb 8, 2012 at 14:27
• -1 because they are very slightly correlated, unless you have data I did not; only downvoting because I took the trouble to check and posted the answer here 16 hours ago and you still didn't take this into account in your reply. Feb 8, 2012 at 16:23
• @Rex: the data you are mentioning is a statistical study based on a fraction of the german population. It is NOT a biology paper that exposes a clear mechanism by which sex determination among siblings is correlated. Counting is not proof, and I think your argument is not robust. Feb 8, 2012 at 22:06
• @SebastienHervieu - A statistical study is better evidence than nothing, which is what you provided. All sorts of things affect the sex ratio, so it's not obvious that there should be no correlation. This is exactly why, as I said, it's more of a biology question than anything else; if you get the biology wrong, you'll start with the wrong priors and not know how large various different effects are. Feb 8, 2012 at 22:29
• @Rex: A statistical study is at best a an educated hint. It is not a proof. It is a poll. And polls usually prove nothing. Your argument is not robust, mine is simpler than yours, but is essentially correct. Feb 9, 2012 at 6:15

You are approached at a party by a shrouded figure who informs you that he/she has a brother. Next he/she asks you to guess his gender. What is the probability that he/she is actually male?

Seems to me that he is male.

• Exactly what I thought. Maybe this question was intended to examine the ability to pay attention to detail and not overlook things based on familiarity of structure. Feb 13, 2012 at 21:28

This is a biology question more than a math or logic (philosophy) question. It turns out that we need to know three biological facts:

1. Are siblings' sex correlated to each other at all?
2. What is the (uncorrelated) sex ratio in humans (males/females)?
3. What is the rate of atypical physiological sexual morphology (hermaphrodite, etc.).

and we need to know two sociological facts:

1. What is the probability that men vs. women will go to a party shrouded and ask random strangers questions about their sex after revealing that they have a brother?
2. What fraction of people identify as some gender that is different than their biological sex?

Ignoring the sociological facts as irrelevant by construction in a thought experiment, we can focus on the biology.

One might assume that the answers are "no" and "1.000", in which case the answer is 0.5.

But it turns out that the sex ratio is greater than 1.0 (~1.05, though it's complicated), so if siblings' sex is not correlated, the answer would be 0.5025 or thereabouts.

Sex among siblings also seems to be very slightly correlated (pdf), raising the probability to about 0.503.

Hermaphroditism is negligible compared to these numbers.

• This is by far the most robust answer. Feb 8, 2012 at 18:23

This question is a variant of the Boy or Girl Paradox, which Wikipedia attributes to Martin Gardner. Wikipedia also writes that "the paradox has frequently stimulated a great deal of controversy"

Wikipedia calls this question a paradox, because:

• On the one hand, the birth of a boy as the first child means nothing for the second child: the expecting parents should still give the probability of their next child being a boy as roughly 0.5 (I know that's not exact, just pretend for now).

• On the other hand, the guy at the bar (or job interview) might look at the problem differently, because the events have already happened: we know that this parent has two children. What is the likelihood that both are boys vs. the likelihood that one is a boy and one is a girl? Put in that way, the likelihood of having two boys is lower than the likelihood of having children of two genders, because there are two possible states of affairs (out of 4) that would result in that outcome: first boy then girl, and first girl then boy.
In other words, the possible combination of genders for two children is: B/B ; B/G ; G/B ; G/G

The actual solution? It depends how the question is asked, or how and when you discover that one of the children is a boy. From the manner quoted here, the question is ambiguous. If the stranger offered this information to you upfront, than from your perspective, is it as if the stranger is offering information regarding a random child, in which case the answer would be 1/2, or specifically the child that is not the stranger him/herself? In the latter case, the answer would be 1/3, because your original sample size is only the families of two children where one of them is a boy, (or, in another version, you specifically ask, "is one of them a boy", not picking a child at random) and we want to know, of those families, what is the probability that the other child is also a boy.

Eliezer Yudkowsky (a relatively well-known AI researcher) has a lovely story about this problem where writes that an incorrect wording of it changed his life. He, and the linked Wikipedia article, both have good explanations to a Baysian analysis of the problem and its solution.

Of course, the questioner probably should have also stipulated that it's a way of asking a probability puzzle, so that there wouldn't be any demographic or sociological factors to consider. That way, you can discuss this as a case where boy:girl ratio is 1:1 and that the sex one child doesn't affect the probability of the sex of a second child (which technically isn't true, partly because there's a possibility that the two children are twins), etc. because after all, that's not really the exciting part of the puzzle, is it?

• A big +1 for the story by Yudkowsky and the elaborate, to-the-point answer. Well done! Aug 7, 2014 at 12:55

I'll answer from the completely math-style logic POV.

We know this person has a brother. What is the probability that s/he's male?

Let us assume that half of the population is male and the other half female, that this is very uniformly distributed, that the gender of previous children doesn't affect gender of any future child of a couple, etc. Let us also assume for a moment that the figure has exactly 1 sibling.

Then a priori, there were four equally likely possibilities in his family were BB, BG, GB, and GG. Now we know one is a boy. So we are left with 3 equally likely possibilities: BB, BG, and GB. In two of the three, the shrouded figure is a female. Correspondingly, 1/3 that he's a guy.

Now suppose the shrouded figure has 2 siblings. Each of the previous possibilities can be modified 'in pairs,' so that we get BBB/BBG, BGB/BGG, GBB/GBG, GGG/GGB. Knowing one is a boy leaves BBB/BBG, BGB/BGG, GBB/GBG, GGB. Thus we have 1 situation in which he's a guy, 3 in which s/he's 50/50, and 3 in which she's a girl. So now the chances are smaller - 5/14 that he's a guy (approximately 1/3).

And so on. In general, it's more likely given these assumptions and that we don't know the relative ages that the figure is a girl. It is Monty Hall style.

• This reasoning is not correct. For the three families BB, BG, and GB, the shrouded figure has twice the chance of being from the BB family, since both members of this family have brothers. Thus the shrouded figure has a 1/2 chance of being male. Feb 8, 2012 at 5:50
• That's what I think - the question was intended to be a Monty-Hall style question but it was asked incorrectly, undermining the main premise. Feb 8, 2012 at 10:14
• @Jim: you are absolutely and positively 100% incorrect. This is exactly the Monty Hall problem in another form and while mixedmath's explanation is a little confusing because of the tangent of having multiple siblings; mixedmath is absolutely correct. Your claim of having twice the chance of being from a BB family makes no sense. If you don't believe the math then simulate this for yourself. Write a program that randomly generates the gender of 2 siblings. If sib 1 is a boy then increment the count for the gender of sib 2. Vice-versa if sib 2 is a boy. Repeat a million times it will converge.
– Dunk
Feb 23, 2012 at 22:56