Coding Horror: The Problem of the Unfinished Game • Hacker News Board

Re: Coding Horror: The Problem of the Unfinished Game

#11

Post by randomwalker » Wed, Dec 31, 2008, 3:01 PM UTC

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."

Re: Coding Horror: The Problem of the Unfinished Game

#12

Post by biohacker42 » Wed, Dec 31, 2008, 3:10 PM UTC

randomwalker wrote: post #11

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'…

That's a good point and there a lot of people who are very good at math, but hate word problems with a passion.

Re: Coding Horror: The Problem of the Unfinished Game

#13

Post by hhm » Wed, Dec 31, 2008, 3:14 PM UTC

randomwalker wrote: post #11

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'…

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.

Re: Coding Horror: The Problem of the Unfinished Game

#14

Post by dant » Wed, Dec 31, 2008, 3:17 PM UTC

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.

Re: Coding Horror: The Problem of the Unfinished Game

#15

Post by brl » Wed, Dec 31, 2008, 3:18 PM UTC

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%

Re: Coding Horror: The Problem of the Unfinished Game

#16

Post by anamax » Wed, Dec 31, 2008, 4:03 PM UTC

byrneseyeview wrote: post #9

euroclydon wrote: post #7

Earlier quoted context omitted.

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...

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).

> 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....

Re: Coding Horror: The Problem of the Unfinished Game

#17

Post by [deleted] » Wed, Dec 31, 2008, 4:19 PM UTC

randomwalker wrote: post #11

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'…

[deleted]

Re: Coding Horror: The Problem of the Unfinished Game

#18

Post by soldarnal » Wed, Dec 31, 2008, 4:20 PM UTC

randomwalker wrote: post #11

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'…

The BG vs GB thing is difficult to understand. But easier to understand is that 50% of two child familes are mixed gender, while 50% are same gender. With 40 families having two children, 20 will have a boy and a girl, and 20 will have either two boys or two girls. Of the same gender familes, 50% (10) will be boy/boy and 50% (10) will be girl/girl. If you learn that a particular family has a daughter, then you know that family isn't one of the 10 families with two boys. This leaves 30 families, 20 of which (or 2/3) have one boy.

Likewise, for some reason the Monty Hall problem is hard to understand in three door form. But if you change the problem to 100 doors, and explain that Monty's assistant closes 98 bad doors, it's easier (at least for me) to see that the original guess had a 1/100 chance of being right, while the other door is 99/100 likely to be the right one.

Re: Coding Horror: The Problem of the Unfinished Game

#19

Post by notauser » Wed, Dec 31, 2008, 4:21 PM UTC

liuliu wrote: post #4

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.

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

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

Re: Coding Horror: The Problem of the Unfinished Game

#20

Post by cedsav » Wed, Dec 31, 2008, 4:43 PM UTC

byrneseyeview wrote: post #9

euroclydon wrote: post #7

Earlier quoted context omitted.

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...

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).

If the problem were "Given an ordered number of events (birth) what is the probability distribution of the last event". Then yes, it would be 50/50.

But you're looking for at least one boy. If you had 1000 children, intuitively, the probability of at least one being a boy is huge.