

Coding Horror: The Problem of the Unfinished Game - twampss
http://www.codinghorror.com/blog/archives/001203.html

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randomwalker
Yeah, this is a famous puzzle. The answer is supposed to be 2/3, because what
the question is asking you to do is consider all the parents in the world
where _at least_ one of the two children is a girl. Then you're left with 3
possibilities, BG, GB and GG.

If you phrase the question like that, everyone will get the right answer. The
reason people get it wrong is that _people don't normally talk like that_.
Imagine you're at a party, and someone tells you they have two kids, and "one
of them is a girl." Clearly, they mean that the other is a boy, which means
the answer is 100%.

But the most intuitive way of interpreting the question is that you know that
a _specific_ child is a girl, say because the person brought one kid to the
party, who turns out to be a girl. With this interpretation, the obvious
answer of 50% is in fact correct.

You often hear the complaint that people don't understand math. In this
instance, however, an equally valid way of explaining what's going on is that
mathematicians don't understand people.

This criticism applies partially to the normal game-show version of the Monty
Hall problem, but I think there the wording is genuinely ambiguous regarding
the host's behavior, and my answer would be "not enough information."

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hhm
The game-show version of the Monty Hall problem isn't ambiguous... people even
had the chance to see the show before knowing the mathematical problem. I
agree there are different ways to state the problem, but the wording doesn't
explain why it confuses people.

~~~
procrastitron
Actually, the strongest criticism that I've heard about the Monty Hall problem
is that the game show didn't actually work like that.

I've tried explaining the problem to both people who've seen the show and
those who haven't. My experience is that if you've seen the show you'll never
accept the question, let alone the answer. Meanwhile, those who never watched
the show can be convinced.

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WilliamLP
Take a random parent with two children. You ask them if at least one is a boy
and they say yes. The probability that the other is a girl is 66.6%

Take another random parent with two children. You ask them if the oldest child
is a boy and they say yes. Now the probability that the other is a girl is
50%.

This confuses people. (The point is the second scenario gives you more
information since it's a subset of the first.)

~~~
thras
Exactly correct. Your comment should be at the top of this discussion. It took
me forever to figure the above out when I first ran into this puzzle in 7th
grade.

And although this problem is often used to show that humans are bad at doing
probability in their heads, note how simple it is to do the probability
calculation for the two scenarios that you've mentioned.

What's actually hard for humans is describing models or simulations, not doing
the probability. Once you explained the possibilities for original problem in
a more detailed way, the answer(s) becomes obvious.

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brl
This is easier to understand if you start by listing all of the possibilities.

    
    
      Two boys:        BB
      Two girls:       GG
      Boy, then girl:  BG
      Girl, then boy:  GB
    

So there are four possibilities: (BB, GG, BG, GB).

If you know that one of the children is a girl, then BB is impossible and you
can remove it from the list. This leaves only 3 possibilities (GG, BG, GB).

This is the set you use to calculate the probability and there are two ways
out of three that the person could have both a boy and a girl: BG and GB

That gives you 2/3 or 67%

~~~
pstuart
There are neither 4 nor 3 possibilities: it was given that one child is a
girl. So it's either GB or GG. 50% FTW.

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liuliu
67%. The percentage of having one boy and one girl is 50%, "one of them is a
girl" eliminate the condition of having two boy.

~~~
notauser
Except that real data seems to disagree with you - the chance is about 50%.

<http://www.in-gender.com/xyu/Odds/Gender_Odds.aspx>

~~~
notauser
A link to an analysis based on an actual study was modded down?

Who are you people and what did you do to HN.

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designtofly
The analysis you linked to does not refute anything said here. In fact, the
conclusions of the study you showed are a necessary assumption for this
problem (that each gender has an equal probability and independent of birth
order). This problem has nothing to do with biology but rather using posterior
information to come up with a probability given a statistical experiment.

However, I did not vote you down.

~~~
dgabriel
But why does it necessarily have nothing to do with biology? It seems you've
added a condition to the question that does not exist. Additional data points
are quite useful, and using them to find the correct "real world," outcome
seems most logical.

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mechanical_fish
+1: The Monty Hall Problem Never Gets Old :)

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dant
Consensus is clearly 66% or 50% depending on whether you think the GB and BG
combinations are the same thing in the context of the question.

But isn't it true that more boys than girls are born (because boys die younger
so evolution tries to balance it out a bit)?

Does anyone know if certain fathers can only produce one sex of child? If so
then having one girl would increase the chances of having another girl
slightly.

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ars
Evolution doesn't try to balance anything - evolution is not a person, and it
doesn't think.

There is a slight difference in the swimming speed of XX vs XY sperm, which
accounts for the difference. Also I believe females have a slightly better
survival ratio, the reasons are complicated, but include the fact that all
fetuses start as female, and then are modified by testosterone to be male,
i.e. female is the default. Plus females have XX so some genes are doubled
which helps them.

It's pretty much impossible for a father to make just one or the other because
of that way it's produced.

The father has XY cells, which split in half to make sperm, one half become a
male sperm the other female. So sperm is always made in pairs.

There can be differences in the mother that affect one or the other
differently though (PH for example, and the sperm are not the same size).

~~~
dant
I wasn't suggesting that evolution thinks, I was suggesting that we've evolved
to produce more males so that there's an optimum 50/50 balance of the number
of males and females at around early adulthood. Considering some of the other
things that evolution has achieved that seems like a pretty reasonable thing
to suggest.

Your explanation of whether certain couples can only produce males or females
was pretty interesting though, thanks.

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euroclydon
Is not each birth an independent random event, and thus the likelihood that
they would have a boy, given that they already have a girl, simply 50%?

~~~
Shamiq
That's the same reasoning I have. P(B|A) = P(B) for independent variables.

~~~
euroclydon
On the other hand, if a couple told you they had five kids, at least four are
boys, you would say to yourself, "it's not very likely that any couple would
have five boys", and you would be correct...

~~~
byrneseyeview
Yes, BBBBB is rare. But BBBBG is just as rare (at least statistically. In
practical terms, I suspect that having four boys in a row would be too
exhausting and irritating, so they'd have given up by then).

~~~
anamax
> Yes, BBBBB is rare. But BBBBG is just as rare

Actually, it isn't. More boys are born than girls. (However, boys are somewhat
more likely to die young.) Also, the odds for a given mating pair are not the
same as the odds for the population as a whole.

And then there's post-conception sex-selection....

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shiranaihito
If it's fifty-fifty between genders, and each birth is an independent event,
but a boy + girl is like umm.. a chain of independent events or something?

My guess is 25% - 0,5 times 0,5..

But what exactly is the right answer? There doesn't seem to be a consensus
yet.

