Mathematical Probability Puzzle

[needimprovement]

Your new mathematics-obsessed friend says to you, "I have two children.

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

 

The first thing you think to ask him is, "What the heck does Tuesday have to do with it?"

"Everything!", he replies...

 

So what is the probability?

[the tree]

6/13

Edited September 23, 2010 by the tree

[Dimdims]

how the hell did you get that answer?

[the tree]

The plausable situations are that he has a daughter born on one of seven days, or a second son born on one of six days - since a second son born a tuesday would contradict the premise. So that is, out of 7+6=13 plausable situations (each with fairly equal probability), 6 that would result in the friend having two sons. So 6/13. Again with the slightly too easy.

[uncool]

P(2 sons | 1 son born on Tuesday) = P(2 sons & 1 born on Tuesday)/P(1 son born on tuesday)

 

= P(1 son born on Tuesday | 2 sons) * P(2 sons)/(P(1 son born on Tuesday | 2 sons) * P(2 sons) + P(1 son born on Tuesday | 1 son) * P(1 son))

 

= ((13/49) * 1/4)/((13/49) * (1/4) + (1/7) * 1/2)

 

= (13 * 1)/(13 * 1 + 7 * 2) = 13/27.

 

Now, this assumes that his actually stating the fact is independent of whether he has 1 son or 2 sons, etc.

=Uncool-