Puzzle: I have two children ...

[bmd]

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

Gary Foshee, a collector and designer of puzzles from Issaquah near Seattle walked to the lectern to present his talk. It consisted of the following three sentences: "I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"

The event was the Gathering for Gardner earlier this year, a convention held every two years in Atlanta, Georgia, uniting mathematicians, magicians and puzzle enthusiasts. The audience was silent as they pondered the question.

"The first thing you think is 'What has Tuesday got to do with it?'" said Foshee, deadpan. "Well, it has everything to do with it." And then he stepped down from the stage.

The gathering is the world's premier celebration of recreational mathematics. Foshee's "boy born on a Tuesday" problem is a gem of the genre: easy to state, understandable to the layperson, yet with a completely counter-intuitive answer that can leave you with a smile on your face for days. If you have two children, and one is a boy, then the probability of having two boys is significantly different if you supply the extra information that the boy was born on a Tuesday. Don't believe me? We'll get to the answer later.

Link to answer.

 

[Johnny Canuck3]

I tried reading that, but I just couldn't. This is why I'm not a doctor or dentist, earning real money. I just can't do math.

I'd have thought the probability was 50/50.

 

[Santino]

When he says 'one is a boy born on a Tuesday', is he implicitly saying that ONLY one is a boy born on Tuesday?

 

[AnnO'Neemus]

Ooh, interesting.

Makes sense but doesn't make sense at the same time.

 

[1927]

Ooh, interesting.

Makes sense but doesn't make sense at the same time.

Innit.

The probability of him having two boys should be constant gievn the same parameters, but it isn't according to that solution.

 

[fractionMan]

I'm trying really hard not to look at the answer.

 

[fractionMan]

How can it not be 50/50? Grrr.

 

[killer b]

How can it not be 50/50? Grrr.

because the second child can't be a boy born on a tuesday, which means (i think) the chances are girl 53.8/46.2 boy

 

[fractionMan]

Nah, you could say "... and the other is also a boy born on a tuesday."

 

[RaverDrew]

This is where mathematical probability is flawed.

It's like when the maths geeks proclaim that the result of the lottery is just as likely to be 1, 2, 3, 4, 5, and 6 as any other combo. But anybody with half a brain knows that in the real world it would NEVER fucking happen.

 

[RaverDrew]

because the second child can't be a boy born on a tuesday, which means (i think) the chances are girl 53.8/46.2 boy

There is nothing to say that the second child can't be a boy born on a tuesday.

 

[Crispy]

The question needs ammending to say only one is a boy born on a tuesday. the question doesn't preclude two tuesday boys.

and yes 123456 is just as likely as any other combination. The bit of your brain going "no no no that's not possible" is the bit of your brain that tells you to play the mugs game in the first place. Fight it!

 

[Santino]

According to my spreadsheet there are 27 combinations of child/day of birth featuring a boy born on Tuesday. 13 have two boys, and 14 have a girl and a boy.

 

[ivebeenhigh]

1,2,3,4,5,6 is statistically chosen by lots more people than any other random collection of numbers, people think they are being clever by saying "its just as likely as any other set of numbers" which is correct. However you are going to share the winning prize with hundreds of people should the jackpot ever come up

 

[strung out]

According to my spreadsheet there are 27 combinations of child/day of birth featuring a boy born on Tuesday. 12 have two boys, and 13 have a girl and a boy.

what do the other two combinations have?

 

[fractionMan]

Maybe he's using "one" to refer to himself. Not that it changes anything.

 

[Santino]

I'm not getting my head around this. Say I get a load of people in who have two children, one of whom is a boy. I divide them into two groups. I ask the first group what day their son was born in. Have I affected the number in each group who will have two boys?

 

[Santino]

what do the other two combinations have?

I don't know what you mean.

 

[quimcunx]

1,2,3,4,5,6 is statistically chosen by lots more people than any other random collection of numbers, people think they are being clever by saying "its just as likely as any other set of numbers" which is correct. However you are going to share the winning prize with hundreds of people should the jackpot ever come up

Money isn't the only prize. Think how smug they will all feel.

 

[beesonthewhatnow]

*scratches head*

B=Boy, G=Girl

So, we can have:

First Child, Second Child

B(Tue), B(Mon)
B(Tue), B(Tue) *
B(Tue), B(Wed)
B(Tue), B(Thur)
B(Tue), B(Fri)
B(Tue), B(Sat)
B(Tue), B(sun)

B(Mon), B(Tue)
B(Tue), B(Tue) *
B(Wed), B(Tue)
B(Thur), B(Tue)
B(Fri), B(Tue)
B(Sat), B(Tue)
B(Sun), B(Tue)

B(Tue), G(Mon)
B(Tue), G(Tue)
B(Tue), G(Wed)
B(Tue), G(Thur)
B(Tue), G(Fri)
B(Tue), G(Sat)
B(Tue), G(sun)

G(Mon), B(Tue)
G(Tue), B(Tue)
G(Wed), B(Tue)
G(Thur), B(Tue)
G(Fri), B(Tue)
G(Sat), B(Tue)
B(Sun), B(Tue)

The * option is repeated, so we have 27 possible combinations of children.

Of those, 13 are both boys.

So the answer is there is a 13/27 chance that he has two boys.

I think.

 

[fogbat]

1,2,3,4,5,6 is statistically chosen by lots more people than any other random collection of numbers, people think they are being clever by saying "its just as likely as any other set of numbers" which is correct. However you are going to share the winning prize with hundreds of people should the jackpot ever come up

Heh. Idiots.

It's why I use 7,8,9,10,11,12

 

[strung out]

*scratches head*

B=Boy, G=Girl

So, we can have:

First Child, Second Child

B(Tue), B(Mon)
B(Tue), B(Tue) *
B(Tue), B(Wed)
B(Tue), B(Thur)
B(Tue), B(Fri)
B(Tue), B(Sat)
B(Tue), B(sun)

B(Mon), B(Tue)
B(Tue), B(Tue) *
B(Wed), B(Tue)
B(Thur), B(Tue)
B(Fri), B(Tue)
B(Sat), B(Tue)
B(Sun), B(Tue)

B(Tue), G(Mon)
B(Tue), G(Tue)
B(Tue), G(Wed)
B(Tue), G(Thur)
B(Tue), G(Fri)
B(Tue), G(Sat)
B(Tue), G(sun)

G(Mon), B(Tue)
G(Tue), B(Tue)
G(Wed), B(Tue)
G(Thur), B(Tue)
G(Fri), B(Tue)
G(Sat), B(Tue)
B(Sun), B(Tue)

The * option is repeated, so we have 27 possible combinations of children.

Of those, 13 are both boys.

So the answer is there is a 13/27 chance that he has two boys.

*applause*

(though santino got there first, with a spreadsheet too!)

 

[fractionMan]

Heh. Idiots.

It's why I use 7,8,9,10,11,12

Me too!

We're so clever

 

[Santino]

Another way of looking at it:

BTu, BMon
BTu, BTu
BTu, BWed
BTu, BThu
BTu, BFri
BTu, BSat
BTu, BSun
BTu, GMon
BTu, GTu
BTu, GWed
BTu, GThu
BTu, GFri
BTu, GSat
BTu, GSun

There are 14 combinations, 7 with a girl and 7 without.

Why does the order mean anything?

 

[fractionMan]

I think a geek of this magnitude has about a 0% probability of having sex, let alone children.

 

[beesonthewhatnow]

Another way of looking at it:

BTu, BMon
BTu, BTu
BTu, BWed
BTu, BThu
BTu, BFri
BTu, BSat
BTu, BSun
BTu, GMon
BTu, GTu
BTu, GWed
BTu, GThu
BTu, GFri
BTu, GSat
BTu, GSun

There are 14 combinations, 7 with a girl and 7 without.

Why does the order mean anything?

But there are 27 combinations, as your spreadsheet, and my typed effort, show. It's details like this that matter in probability...

 

[strung out]

i'm surprised kabbes didn't turn up

 

[Santino]

i'm surprised kabbes didn't turn up

The longer the thread gets the probability of kabbes making an appearance tends to 1.

 

[Santino]

Where are you, kabbes?

 

[kabbes]

I actually had useful work-related things to do this morning. Extraordinary, I know. But I do occassionally have to make a token appearance. So I didn't spot it until now.

I'll be busy this afternoon too.

Anyway, this is a favourite stats question of old. In fact, it is so well known that it is called the "boy-girl paradox". The day of the week thing just makes it more complicated but the principle is the same.

Firstly, just consider the boys and girls. Then you'll get the principle.

Two children can be BB, BG, GB and GG. So if I say that the eldest is a boy then you are left with BB and BG, which means that the younger is 50/50 for boy or girl.

But two children given only that you know that one is a boy can only be BB, BG and GB. Two of those have a girl as the second option, meaning that the probability that the other child is a girl is 2/3 (and its complement, which is two boys, is 1/3).

This question just takes the same approach and convolutes it with an equivalent question involving days of the week. It appears much more complicated because you have two dimensions, but it is the same principle at work.