The Tuesday boy problem, in under 300 words

http://mikeschiraldi.blogspot.ca/2011/11/tuesday-boy-problem-in-under-300-words.html

by colinprince • 13 years ago

33 62 13 years ago

mikeschiraldi.blogspot.ca

ggreer3• 13 years ago

This is incorrect. There's no reason to assume that the stranger will say, "at least one of them is a girl" only in the case of GG.

In the case of BB, the stranger will always say, "At least one is a boy." Given GB or BG, they have even odds of saying "at least one is a boy" vs "at least one is a girl." Given GG, the stranger will always say, "At least one is a girl." The correct diagram would be this: http://abughrai.be/pics/boy_probability.png

For the probabilities in that post to apply, the stranger would say, "I have two children." Then you ask, "Is at least one of them a boy?" and they answer, "Yes."

Edit: Of course, the answer to, "Given there are two siblings and one is a boy, what are the chances both are boys?" is 1/3. But the phrasing of the puzzle does not reflect that question.

javert4• 13 years ago

The post is not about the probability of what a person with a given pair of children will say.

Rather, what they said is given, and the question is: what pair of children do they have?

You are either going over my head or under my head, so this will probably make one of us temporarily look a bit foolish :).

ggreer3• 13 years ago

The blog post[1] that the author based his short explanation on goes into this. Peter Winkler, a math professor at Dartmouth, is quoted in it[2]:

Suppose the puzzle is phrased this way: X says “I have two children and at least one is a boy.” What is the probability that the other is a boy?

Put this way, the puzzle is highly ambiguous. Computer scientists, cryptologists and others who must deal carefully with message-passing know that what counts is not what a person says (even if she is known never to lie) but under what circumstances would she have said it.

The way the puzzle is phrased, the only way to get 1/3 as an answer is to assume that parents with one girl will never say, "At least one of them is a girl."

1. http://scienceblogs.com/evolutionblog/2011/11/08/the-tuesday...

2. http://blog.tanyakhovanova.com/?p=234

Confusion• 13 years ago

Fine, then I pose to you the same problem description, with the added fact that the stranger is proud of having a boy, will only draw attention to a boy and will thus say "at least one is a boy", independent of whether the other is a girl, while he would never say "at least one is a boy".

You can't solve a logic puzzle by arguing the psychology of a persona depicted in the puzzle.

stephencanon• 13 years ago

You can’t solve a logic puzzle that way, but you can explain the pragmatics of why the “logical” answer is contrary to our expectations based on interaction with real people.

glenngillen• 13 years ago

Afaict tell the initial range of possibilities is:

GG GB BB

The relative ages of the children play no part, they could be twins for all I care. So the initial probabilty of being BB is 1/3. Knowing one of them is a boy then removes GG as an option, and the probability for BB unsurprisingly becomes 1/2.

What am I misunderstanding in this problem?

aytekin• 13 years ago

It matters if the assumption is that possibility of each child being boy/girl is 50%.

The first child can be boy or girl. The second child can also be boy or girl. So the probabilities are like this: BB: 25% GG: 25% BG: 50%

You can replace older/younger on the article with child 1/child 2.

te• 13 years ago

That GB is twice as likely as either GG or BB. For this reason, reference is made to BG to make this more apparent.

gametheoretic11• 13 years ago

You only misunderstand because the author is presenting it stupidly (as you state, incorporating order where order is irrelevant), leading to his own stupid conclusion. Read my post elsewhere in the thread for the full explanation, and if you find this kind of thing interesting, look up the Monty Hall problem (the point of which is that changing the information changes the odds).

acchow• 13 years ago

This explanation ignores the two perspectives of probability - assertions on states of the world (the "frequentist" view) vs. states of the mind (the "bayesian" view).

klodolph4• 13 years ago

It has nothing to do with frequentist versus bayesian statistics this is really just a pure probability theory problem.

ggreer3• 13 years ago

This exact puzzle is discussed in http://lesswrong.com/lw/ul/my_bayesian_enlightenment/:

I responded—note that this was completely spontaneous—"What on Earth do you mean? You can't avoid assigning a probability to the mathematician making one statement or another. You're just assuming the probability is 1, and that's unjustified."

While the equations of probability theory are agreed upon, there are multiple interpretations[1]. In some ways, it's like QM.[2]

1. http://en.wikipedia.org/wiki/Probability_interpretations#Phi...

2. http://en.wikipedia.org/wiki/Interpretations_of_quantum_mech...

javert4• 13 years ago

As far as I can tell, that blog post contains a "bait and switch." Here it is:

But suppose that instead you had asked, "Is your eldest child a boy?" and the mathematician had answered "Yes." Then the probability of the mathematician having two boys would be 1/2. Since the eldest child is a boy, and the younger child can be anything it pleases.

No, it cannot be "anything it pleases." That is the whole point of the blog post that we are all commenting about here on HN, which is re-told on the blog post you are linking to.

Since the post you are linking to is misrepresenting one side of the "argument," I strongly suggest that there is no actual apparentl contradiction here that needs to be resolved.

Yes, you can talk about the actual probability of something vs. making the best possible prediction you have. A good example raised in this post is the point about a biased coin; you know it is biased but not which way, so you assign odds of .5 for heads and .5 for tails. Fine. This is not some massive philosophical conundrum.

The fact is that actual knowledge is neither purely "out there" (e.g. "in the cards") nor purely "in here" (in the mind). This is the classic rationalism vs. empiricism debate, which was settled by Ayn Rand's theory of objective knowledge, where she answers that knowledge is in man's mind, but it is knowledge of external reality. So, "both."

Not understanding this, and other basic epistemological errors, seem to plague lesswrong.com. It is embarassing to see them supposedly championing "reason" under a non-objective epistemology. They come to bizarre conclusions, and that is, at least partially, why.

tarblog• 13 years ago

Ayn Rand settled the age-old epistemology debate? I don't think so.

gametheoretic11• 13 years ago

This is wrong.

In oversimplifying your table for potential idiot readers by introducing the silent variable of older vs younger child, you have introduced order into a statistics problem wherein order is irrelevant and masked the fact that what matters is which of the children you already know to be a boy. You grant this difference for the BG pairing, thus producing two options, but not for the BB pairing. There is not ONE way of knowing this for BB, as you present in the table, but two: you know the older is the boy, or you know the younger is the boy. 2/4 = 1/2. QED.

What you should have done in the first place, however (unless your aim is to produce a blog post which, apparently, can convince otherwise intelligent people that irrelevant information can magically become relevant) was simply remove the one boy from the equation and reformulate the question. What is the probability that this other child is a boy (and thus that both are boys)? 1/2. Fuck the table. Had you not used a table, this would never have happened. But it feels authoritative, right? Thank you for the psychology lesson.

Downvotes but not refutations, because there aren't any. I'll assume it's my tone. My opinion of the competence of the average reader on this site has plummeted reading the other comments, however, so hey, fuck you guys too. :)

klodolph4• 13 years ago

You're oversimplifying.

The problem is the interpretation of the question in precise terms. We are given some background information and then told the observation of a particular random variable. In the given article, the random variable is interpreted as "yes/no: at least one child is a boy born on a Tuesday". Rambling about "order" does not change the fact that the article is correct, given the definition of the random variable that is observed.

The problem is that this interpretation is only correct if you imagine that you are always given the observation of the same random variable no matter what the genders or birthdays of the children actually are. For example, if I had two girls born on Wednesday, I would then tell you "no child is a boy born on a Tuesday". That's an unlikely scenario which is what makes it legitimately surprising.

It's like the Monty Hall problem. What if the host picks the door in advance? What if the host only opens a door with a goat under the condition that you picked the door with the prize? What if the host only opens a door with a goat under the condition that you picked the wrong door?

The article has problems but "wrong" is not one of them.

so hey, fuck you guys too

Hacker News is a terrible place to visit if you want to take the voting system personally. Don't make this about you. It's about the post.

gametheoretic11• 13 years ago

I'm not, you're overcomplicating! X is a boy. What is the chance that Y is also a boy? 1/2 or 1/3? Non-rhetorical, answer the question as posed.

Sandman2• 13 years ago

Wait, explain it to me like I'm 10 years old.

As I see it, there are three options - either it's a boy-boy pair, a girl-girl pair or a boy-girl pair. Now, since OP introduced the age variable, let's give the kids names to make things easier to understand. Let's say that the boys are named John and Jack, the girls Jill and Judy and if it's a boy-girl pair then they're named Jack and Jill.

Now if it's a boy-girl pair, we have exactly two options: either Jack is older than Jill or Jill is older than Jack. This is shown in the first picture.

However, we also have two options for boy-boy and girl-girl pairs: Either Jack is older than John or John is older than Jack. Either Jill is older than Judy or Judy is older than Jill. This is not shown.

Since we know that one of them is a boy, we can dismiss both girl-girl pairs. Now we're left with four options. In 50% of these options the other child is a boy.

What am I missing?

ColinWright9• 13 years ago

Your reasoning works if the person says "The older is a boy." You claim that this is correct.

Consider ...

Assuming the sex of a child is 50:50, then consider all couples with exactly two children. Of those, consider only those who can truthfully say "at least one of my children is a boy." Of those couples, 1/3 will have two boys.

Interpreting the initial problem in this manner, the answer of 1/3 is right. What I'd like to know is why you think the initial problem should be interpreted in any different way.

brazzy4• 13 years ago

You are an overconfident ignoramus. Thanks for demonstrating the Dunning-Kruger effect so impressively.

The reason for having BG and GB is the fact that it's twice as likely to occur than each of GG and BB.

gametheoretic11• 13 years ago

Hahahaha oh my god what a response. The fact that you think you're pointing anything out to anyone with that second paragraph is just priceless.

In GB and BG, he has only one way of telling you which is the boy. In BB, he has two. Thus, four possibilities a) GB, b) BG, c) BB and he gave you the first, d) BB and he gave you the second.

Your brain will of course refuse to process the above due to your earlier remark and the sweet, sweet motherfucking irony it would mean.

ajanuary4• 13 years ago

I did an experiment: I kept flipping two coins [1]. If it came up tails, tails I ignored it. If it came up heads, heads I noted it down in one column. If it came up heads, tails I noted it down in another column. After many flips the ratio was 1:3.

As far as I can tell this is an accurate simulation of the problem. We have two things, coins or children, which have a 50:50 chance of being one value or another. We are told one combination of values isn't the case, so we discount them from the simulation. We then use the frequency of the target combination to approximate its probability. We haven't added any artificial ordering into the problem.

I know it's not well argued logical reasoning, but it supports the articles approach. Can you point out where I tripped up in the simulation?

[1] Of course I didn't actually sit there flipping coins. I wrote a program to do it. (let [sample-size 10000000] (/ (get (frequencies (take sample-size (filter #(not= % [:girl :girl]) (repeatedly #(vector (rand-nth [:boy :girl]) (rand-nth [:boy :girl])))))) [:boy :boy]) sample-size))

icameron• 13 years ago

Instead of actually programming, I flipped 2 coins this morning and left them on my desk. They are still sitting there. Either HT, TH, HH or TT

I just remembered one of them is tails, so is there is a 1/2 chance that they are both tails now? Nope it's 1/3 chance they are both tails. Either HT, TH, or TT.

gametheoretic11• 13 years ago

We haven't added any artificial ordering into the problem.

Yes you have! To run the experiment, you have to treat TH and HT as different but you "pretend" not to do so by adding them into the same pile. You're pretending to know that the father tells you which child is the "known boy", but by the statement of the problem, you don't know that. Why is this an error? Because the father must follow a rule - he must give you a boy, and there are two ways for him to do that in BB but you only recognize one. If you go the "table-minus" route, which both the author and your experiment do. But you can't do that without affecting the odds! By telling you they aren't both girls, he lowers the chance that either of them might be. Yes, I'm serious.

Correct formulation of table:

a) BG; b) GB; c) BB and he gives you the first; d) BB and he gives you the second.

archgoon• 13 years ago

The original article is more interesting and sheds light on the hidden assumptions that go into the calculation. It also considers under which sets of assumptions the answer is not 1/2.

http://scienceblogs.com/evolutionblog/2011/11/08/the-tuesday...

The attempt at shortening it down to under 300 words omits all the complications and subtlety involved in the problem.

willvarfar• 13 years ago

Now, we should mention at this point that we are discussing mathematical children who have no existence outside the world of probabilistic brainteasers. So we’re not worrying about the fact that a disproportionate number of children are born on Mondays and Tuesdays, since C-sections aren’t usually scheduled for the weekends. We’re also not worrying about twins or triplets. Or any other sort of “real world” consideration that might occur to you.

The one that occurred to me is that the probability of getting a boy or girl is not 50%, and the probability of getting two same-gender children is not 50% either. Here are some nice real stats: http://www.in-gender.com/XYU/Odds/Gender_Odds.aspx

kevinjones• 13 years ago

I also thought the original article was better, and clearer. As you say—it's the hidden assumptions which are interesting.

In maths exams at school the problems are always presented in the form of: "a random family was chosen from ...". (Similar to "a smooth cube is resting on a smooth surface".

In real life, it's more difficult. The original article looks at real life, whereas the 300-word version has reduced it to the exam version. Of course it's "easier" to understand the exam version, but this wasn't the point.

valtron• 13 years ago

"I have a two children, one of which is a boy. What's the probability the other is a boy?"

Intuition wrongly treats the possibilities as sets and assigns probabilities 1/2 BB, 1/2 GB, rather than 1/3 BB, 1/3 GB, 1/3 BG.

hislaziness• 13 years ago

What if they are twins? The explanation does not cover that and it sure changes the probabilities.

klodolph4• 13 years ago

If we're going to bring reality into this, I think we have to take into account the fact that only very weird people tell you that "at least one child is a boy born on a Tuesday" and we have to consider the possibility that they are lying in order to test your knowledge of probability.

hudibras• 13 years ago

And if we're really going to bring reality into this, then we'll need to consider the ages of the children, because of the at-birth sex ratio (105 newborn boys for every 100 newborn girls[0]) and differing mortality rates for each sex.

[0]http://en.wikipedia.org/wiki/Human_sex_ratio

chris_wot• 13 years ago

I think it's probably good to be precise in this case. It should state that "at least one child is a boy born on a Tuesday, but one is older than the other by at least a day".

millstone• 13 years ago

What I dislike about this question is the flavor text. It seems designed to mislead the reader by invoking their “realism node,” and then requiring them to suspend it.

From archgoon’s link:

"This puzzle confounds people legitimately, however, because most of the ways in which you are likely to find out that X has at least one boy contain an implicit bias which changes the answer.”

Exactly right. The phrasing in the problem asks the reader to consider a particular scenario, not isolated facts. Consider what happens if you take the question at its word:

If a person on the street says to you, "I have two children," what is the probability that they're both boys?

Hell if I know. Maybe one is a boy and the other is a grown man? Or maybe he has five children, BBGGG. “I have two children, both of which are boys” is technically correct if there’s two boys and three girls, right? Or maybe he’s just outright lying! How do I assign a probability to any of those?

Consider this phrasing instead:

“If you flip a coin twice, and the first and/or the second is heads, what’s the probability that both are heads?” I’d expect more people to get this right than the boy/girl question.

Also, as an aside, punctuation is important:

“I have two children. At least, one of them is a boy.”

(Yikes!)

heyitsnick• 13 years ago

"I met a man on the street, he said he had two children. I asked him, is at least one of them a boy? He said yes. What's the probability both of them are boys?"

This removes the whole ambiguity of why he would assert "at least one of them is a boy," but still makes the question a bit tricky/counter-intuitive, which is useful for teaching the quirks of probability.

zamalek• 13 years ago

What occurred to me is that as more conditions are added to the question the answer seems to tend toward the naive assumption of 1/2 (in his explanation 0.25, 0.33, 0.42, 0.48). This is like a strange cross-over between word problems, probability and unbelievably calculus. Essentially:

lim[generic -> specific] f(P("is one child of two a boy")) = 1/2

Obviously I have no proof for it but it is an interesting observation.

The Tuesday boy problem, in under 300 words

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