Boys and Girls

[Gavin Chipper]

Two questions - the first is quite old and may even have come up on here before but the second I only heard recently.

1. I have two children. One is a boy. What is the probability I have two boys?

2. I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?

Hopefully this will lead to some interesting discussions.

[Lesley Hines]

Since the two events are unrelated, it should be 50:50 for both. However, the female survival rate is slightly higher than that of boys (largely, as I understand it, as girls have XX the duplicate X helps counteract some birth defects which is may be highlighted in boys) so it's about 49:51 boys:girls. Ish. For both questions. That's my logic, anyway

[Charlie Reams]

And I thought the plane-on-a-treadmill thread was gonna be bad...

[Jon Corby]

1. 1/3.
2. 13/27.

[Hugh Binnie]

I get the same stupid answers as John.

What if you have two children; one a blond haired, blue eyed boy born on the 26th May. What are the chances of having two sons in that case?

[Howard Somerset]

Agree with Jon for Q1.

As for Q2, I don't see what relevance Tuesday has as all. So I go for 1/3 again.

[Jon Corby]

I get the same stupid answers as John.

What if you have two children; one a blond haired, blue eyed boy born on the 26th May. What are the chances of having two sons in that case?

Well, I assumed that girl/boy was equally possible, and that births on all seven days of the week were equally possible, and just did the counting from that. These were fairly straightforward 1/2 and 1/7 chances. You'd have to come up with similar figures for blond haired (how many hair colours are you counting - what counts as blond exactly?) and likewise eyes.

[Bob De Caux]

Agree with Jon for 1 and 2.

I think rather than expecting an answer (as you would need more info), Hugh's point is that the more specific information you add, the closer the probability gets to 1/2.

[Hugh Binnie]

Well, I assumed that girl/boy was equally possible, and that births on all seven days of the week were equally possible, and just did the counting from that. These were fairly straightforward 1/2 and 1/7 chances. You'd have to come up with similar figures for blond haired (how many hair colours are you counting - what counts as blond exactly?) and likewise eyes.

Well if you just let hair colour be blond or not blond, and eyes blue or not blue, each equally likely, and assume all birthdays are equally likely (ignoring the problematic 29th February) then the odds of having two boys would be 10219/20439, it seems (if my maths is 'right'). But this is silly — you can't just introduce lots of extraneous details & increase the odds of having two boys.

[Jon Corby]

But this is silly — you can't just introduce lots of extraneous details & increase the odds of having two boys.

Erm... that makes perfect sense to me When you absolutely identify one boy (ie a particular being) as being "the one" that you mention, the chance is 50%. Therefore the more information you give that distinguishes it to one person (kinda like a game of "Who's Who" ), the more the probability tends towards 50%

[Bob De Caux]

But this is silly — you can't just introduce lots of extraneous details & increase the odds of having two boys.

You can! (well, up to 50% anyway). The more extraneous details you introduce, the more unlikely it is that both boys in the BB scenario satisfy them, as you can see from the answers so far.

One is a boy = 1/3
One is a boy born on Tuesday = 13/27
One is a boy born on a certain date with all your hair specs = 10219/20439

If you could uniquely identify one boy exactly (so that no other boy could have those characteristics), then whether the other one is a boy or girl just reverts to Lesley's independence answer, i.e. 50%

[Howard Somerset]

And all nicely discussed here, even to the extent that Tuesday was the day of the week chosen, only a couple of days ago.

I will probably introduce Q1 to some of my S1 students, but there's no way that I'm going to mention Q2. Well not before they take their exam.

[Hugh Binnie]

If you could uniquely identify one boy exactly (so that no other boy could have those characteristics), then whether the other one is a boy or girl just reverts to Lesley's independence answer, i.e. 50%

Ah yes, of course.

[Howard Somerset]

If you could uniquely identify one boy exactly (so that no other boy could have those characteristics), then whether the other one is a boy or girl just reverts to Lesley's independence answer, i.e. 50%

Very convincing.

[Jon Corby]

And all nicely discussed here, even to the extent that Tuesday was the day of the week chosen, only a couple of days ago.

That link also tells us that Martin Gardner died on Saturday. I thought he'd been quiet lately.

[Gavin Chipper]

One way of looking at it is that you can divide boys not born on a Tuesday, girls and boys born on a Tuesday into three distinct categories. So where it might have been 1/3 before, the fact that you have a boy born on a Tuesday in no ways impacts upon the boys not born on a Tuesday category, so it will get closer to 1/2 the smaller the third category is.

Of course, as Charlie hinted at with the aeroplane comment, it does depend on how this information comes about. But the 1/3 and 13/27 work if you know someone has two children and you ask them outright "have you got a boy?" or "have you got a boy born on a Tuesday" What about this case:

Alan and David are in a shop. Alan tells David that he has two children. David asks him if he has a boy. Yes. David then asks what day of the week was a boy born on - if there are two boys on different days, just pick one at random. Tuesday. What is the probability that Alan has two boys?

Brian walks in. Alan tells Brian that he has two children. Brian asks if he has a boy born on a Tuesday. Yes.

I've been sitting there watching all this. For me, what is the probability that Alan has two boys? Does the order of events matter?

Brian and David discuss their own conversations with each other. What is the probability that Alan has two boys for each of them now?

[Gavin Chipper]

I hadn't worked out the answers to my extra questions when I put them, but I'm going to work through them now. This is my intuitions: when David first questions Alan, and it turns out that he has a boy there's a 1/3 chance that he has two boys. When Tuesday comes up, it's just any day so it makes no difference. Still 1 in 3.

Then Brian walks in and as far as I'm concerned he's just made a lucky guess. So for me it's still 1 in 3. And for David. When they discuss matters, I'd hope that they'd end up on the same answer as David and I learn nothing new so still 1 in 3. Brian does learn something new in this case, because hearing David's story decreases the chances that Alan also has a boy born on a day other than a Tuesday. This would also happen if they came into the shop the other way round. So with a bit more maths:

There are still 27 possibilities for the birth order. If child 1 is a boy born on a Tuesday then child 2 can be any of the other 14 possibilities. If child 2 is a boy born on a Tuesday then child 1 can be any of the 14. That would make 28 except we have included both being Tuesday boys twice. So 27. As it was in the original question.

But now we know that Alan picked Tuesday. Of the 27 possibilities there are 12 where there is a boy not born on a Tuesday, 1 where both are boys born on Tuesdays and 14 where one is a girl.

There is a 14/27 chance that he will say Tuesday and the other child is a girl. 1/27 + 6/27 chance that he will say Tuesday and there's two boys and 6/27 chance that there's two boys and he will not say Tuesday. He says Tuesday. It's now 14/21 for a girl and 7/21 for a boy. 1/3 for a boy. Yes!

[Gavin Chipper]

Also, another intuitive way of looking at it - having a boy born on a Tuesday is fairly "rare" and its chances are increased by having two boys. So if you know someone's got a boy born on a Tuesday, they're more likely than usual to have two boys.

[Alice Moore]

Wow. Totally counter-intuitive. I love it.

[David Roe]

Also, another intuitive way of looking at it - having a boy born on a Tuesday is fairly "rare" and its chances are increased by having two boys. So if you know someone's got a boy born on a Tuesday, they're more likely than usual to have two boys.

Fairly limitedly counter-intuitive, though. It only works if you know nothing at all about the children except that one, you don't know which, was a boy born on a Tuesday. If you have any prior information at all about the Tuesday boy, such as he's the elder, or is called Fred, or has darker hair, then the odds revert to 1/2.

[Kirk Bevins]

Wow. Totally counter-intuitive. I love it.

I think it's a load of bollocks. How can one be 1/3 and one be 13/27? There is clearly an error here. I've read that article which just highlights why I (and so many others) struggle with probability. It makes very little sense. If I want to find out in the World how many families consist of 2 boys, given that one is a boy, I work out that I have a 1 in 3 chance. But that boy was born on a Tuesday...oh, so it's a 1 in 2 chance almost? Rubbish.

[Charlie Reams]

Wow. Totally counter-intuitive. I love it.

I think it's a load of bollocks. How can one be 1/3 and one be 13/27? There is clearly an error here. I've read that article which just highlights why I (and so many others) struggle with probability. It makes very little sense. If I want to find out in the World how many families consist of 2 boys, given that one is a boy, I work out that I have a 1 in 3 chance. But that boy was born on a Tuesday...oh, so it's a 1 in 2 chance almost? Rubbish.

Are you saying that the answer is wrong?

[Michael Wallace]

Wow. Totally counter-intuitive. I love it.

I think it's a load of bollocks. How can one be 1/3 and one be 13/27? There is clearly an error here. I've read that article which just highlights why I (and so many others) struggle with probability. It makes very little sense. If I want to find out in the World how many families consist of 2 boys, given that one is a boy, I work out that I have a 1 in 3 chance. But that boy was born on a Tuesday...oh, so it's a 1 in 2 chance almost? Rubbish.

This might help. In particular, the discussion under the heading 'second question'.

Edit: Basically it boils down to what assumptions you're making, which is usually the case with these tricksy probability problems.

[Jon Corby]

Wow. Totally counter-intuitive. I love it.

I think it's a load of bollocks. How can one be 1/3 and one be 13/27? There is clearly an error here. I've read that article which just highlights why I (and so many others) struggle with probability. It makes very little sense. If I want to find out in the World how many families consist of 2 boys, given that one is a boy, I work out that I have a 1 in 3 chance. But that boy was born on a Tuesday...oh, so it's a 1 in 2 chance almost? Rubbish.

It's not rubbish though. The nuance is in the wording of the question, and understanding that there is no "that boy" - it's the sort of 'double-counting' involved that when you say "one is a boy" what you mean is "[at least] one is a boy [and I'm not referring to either of the boys specifically if both are boys]" that gives you the answer that you don't find sensible.

Consider this similar example: Take a pack of cards and deal a bridge hand (13 cards).

a) If it contains an ace, what is the probability that it contains more than one ace?
b) If it contains the ace of spades, what is the probability that it contains more than one ace?

b) is noticeably higher than a). Does the ace of spades magically attract other aces? Of course not.

Probability is awesome because (in instances like this) it is completely testable. You don't have to take our word for it. You don't like an answer? Double check it by counting all the possibilities (correctly!) or better still running a simulation. Get a list of all the 2-child families, and you'll see that [roughly] 1/3 of them that contain a boy contain 2 boys, and 13/27 of them that contain a boy born on Tuesday will contain 2 boys. Lovely.

[Howard Somerset]

I think it's brilliant. Was even driving on the motorway this morning, and all I was thinking about was this boy born on thursday question. Looks totally implausable at first (when I first read this thread a day or so ago), but by the time I got home, having done the various calulations in my head, the idea that the two probabilities as mentioned in the OP should even be the same now seems implausable.

This is another that I'm probably going to bore people to death with at future family gatherings.

[Dinos Sfyris]

I think it's brilliant. Was even driving on the motorway this morning, and all I was thinking about was this boy born on thursday question. Looks totally implausable at first (when I first read this thread a day or so ago), but by the time I got home, having done the various calulations in my head, the idea that the two probabilities as mentioned in the OP should even be the same now seems implausable.

This is another that I'm probably going to bore people to death with at future family gatherings.

Make sure to keep your mind on the road, Howard! I have this mental imagery now of you causing a massive motorway pile-up and then when the police come to question you about it you fascinate them with "Did you know that if there are 2 children and one of them is a boy, then there is a 1/3 probability that they are both boys!"

[Howard Somerset]

I think it's brilliant. Was even driving on the motorway this morning, and all I was thinking about was this boy born on thursday question. Looks totally implausable at first (when I first read this thread a day or so ago), but by the time I got home, having done the various calulations in my head, the idea that the two probabilities as mentioned in the OP should even be the same now seems implausable.

This is another that I'm probably going to bore people to death with at future family gatherings.

Make sure to keep your mind on the road, Howard! I have this mental imagery now of you causing a massive motorway pile-up and then when the police come to question you about it you fascinate them with "Did you know that if there are 2 children and one of them is a boy, then there is a 1/3 probability that they are both boys!"

There's no way I'd say the easy one, Dinos. I'd bring in the Tuesday too, with the 13/27 probability.

[Howard Somerset]

This same puzzle has just been mentioned on Radio 4, towards the end of the programme which finishes at 2 pm.