Mathematical Probability Puzzle

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?

Re: Puzzle

very difficult. this is a logical puzzle, very interesting

Just to clarify, do you mean "one and only one is a boy born on Tuesday"?

And we will assume that boys are born 50% of the time (when in reality it is typically more like 53% I believe).

I'm getting 6/13 for a quick answer (if exactly one is a boy). If both could be boys born on Tuesday, then I get close to 64% chance of two boys. But I need to think about that some more before I will commit to those answers.

50%

Since the context is past tense, one child is known to be a boy. The options are:

boy boy
boy girl

The probability of two boys is 1/3. Please don't ask me how. I don't have time right now to go through the process again.

I was stuck between 14/28 and 13/27 but I'm convinced it's 13/27 now.

I changed my mind ...


For either child, that child could be
G) a girl (1/2)
BT) a boy born on Tuesday (1/2 * 1/7 = 1/14)
BN) a boy born NOT on Tuesday (1/2 * 6/7 = 6/14)

If we reguire exactly one child is (BT), the options are
(BT)*(BN) = 6/196
(BN)*(BT) = 6/196
(BT)*(G) = 1/14 = 14/196
(G)(*BT) = 1/14 = 14/196
where the order listed is the birth order.

These are all the families that have exactly one BT. So if we looked at 196 families, there would be 6+6+14+14 = 40 with one BT. Of these 40 families, 12 have 2 boys = 12/40 = 3/10 = 30%

If you include (BT)(BT) with two boys born on Tuesday, there is one more out of 196. So there are 41 possible families, of which 13 have at least 1 BT. Which is 13/41 = 31.7%

My reasoning is as follows:

The "Tuesday Boy" can be either the first or second child.

So the other child can be either a boy or a girl born on any day of the week. So the availabel combinations are(bTu = Boy Tuesday, gMo = Girl Monday etc):


So there are 28 combinations above which result in 14 boys and 14 girls.

However the combination bTu is repeated so should be removed resulting in 27 total combinations 13 of which are boys and 14 girls, i.e. 13/27