
The Tuesday boy problem, in under 300 words - colinprince
http://mikeschiraldi.blogspot.ca/2011/11/tuesday-boy-problem-in-under-300-words.html
======
archgoon
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...](http://scienceblogs.com/evolutionblog/2011/11/08/the-tuesday-
birthday-problem/)

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

~~~
willvarfar
> 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](http://www.in-
gender.com/XYU/Odds/Gender_Odds.aspx)

------
millstone
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
"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.

------
ggreer
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](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.

~~~
Confusion
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
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
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
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.

------
zamalek
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.

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

~~~
klodolph
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
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](http://en.wikipedia.org/wiki/Human_sex_ratio)

------
valtron
"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.

------
acchow
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).

See also
[http://lesswrong.com/lw/oj/probability_is_in_the_mind/](http://lesswrong.com/lw/oj/probability_is_in_the_mind/)

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

~~~
javert
Well, there are only five reasonable ways to disagree.

(1) Do you hold that concepts and principles are purely "in" the referents?

(2) Do you hold that they are purely "in" man's mind?

(3) Do you hold that they are "in" man's mind, but refer to properties of the
referents? (Rand's conclusion)

(4) Do you hold that Rand's argument and conclusion is correct, but somebody
else made it earlier?

(5) Do you hold that there is insufficient evidence to answer the question
(which then raises questions about why you would even engage in an
intellectual discussion)? For example, you could "argue" for a subjectivist
view of knowledge.

An actual answer along the lines of 1-5 would be useful, but stating "I don't
think so" does not provide any reasoning, and thus is utterly devoid of
intellectual content.

You seem to be dismissing a theory without considering and responding to its
evidence, which is fraudulent science. Nobody wants to see that kind of
"argument," even people who nonetheless do agree with the conclusion.

I do think you have the intellectual right to merely state that you disagree,
but you have to say that you don't intend to defend it, or let yourself be
subject to having it pointed out by someone, as I am.

That is because too many people today hide behind the assumption that
agreement or disagreement without evidence presented is enough to establish or
disestablish truth. As I said before, that is fraudulent science.

------
gametheoretic
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. :)

~~~
ajanuary
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
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.

~~~
ajanuary
Of course that's a much simpler way to simulate the problem, but then you get
people trying to argue that HT and TH should be collapsed into a single state.

By using a simulation with repetition it's easier to pose the states as 'TT'
and 'not TT' and use frequencies to show 'not TT' occurs more frequently than
'TT'. I find this starts to help convince people that the states shouldn't be
collapsed, even if it doesn't explicitly explain why.

