> I have two children. One or more is a son. Exactly one of them was born on a Tuesday.
I'm not sure that's what you are supposed to infer.
Looking at your earlier statement:
> I have two children. Here is some information about one of them: son, born on Tuesday.
There are two ways to interpret this.
(1) I am a man pulled at random from the set of [families with two children of indeterminate gender]. Here is some information about one of them: son, born on Tuesday.
(2) I am a man pulled at random from the set of [families with two children of indeterminate gender, one of whom was born on a Tuesday]. Here is some information about one of them: son, born on Tuesday.
We're not selecting from the same initial set in each case - set (2) is more restrictive. A difference in probability is maybe not surprising.
Still, I totally agree with you that it's a bit of a jump to conclude that the man is referring to scenario (2) in which the birthday information is used to narrow the initial set while the gender information is used to determine the probability. Just seems like a trick question to me.
To expand on the numbers a bit more... In scenario (2) the possible combinations are:
(Combo A) G G (0/49 at least one boy TB)
(Combo B) B G (7/49 at least one boy TB)
(Combo C) G B (7/49 at least one boy TB)
(Combo D) B B (13/49 at least one boy TB)
If the man has one boy, then combo A does not apply and the probabilty he has two boys must be 13 / (7 + 7 + 13) == 13/27.
I'm not sure that's what you are supposed to infer.
Looking at your earlier statement:
> I have two children. Here is some information about one of them: son, born on Tuesday.
There are two ways to interpret this.
(1) I am a man pulled at random from the set of [families with two children of indeterminate gender]. Here is some information about one of them: son, born on Tuesday.
(2) I am a man pulled at random from the set of [families with two children of indeterminate gender, one of whom was born on a Tuesday]. Here is some information about one of them: son, born on Tuesday.
We're not selecting from the same initial set in each case - set (2) is more restrictive. A difference in probability is maybe not surprising.
Still, I totally agree with you that it's a bit of a jump to conclude that the man is referring to scenario (2) in which the birthday information is used to narrow the initial set while the gender information is used to determine the probability. Just seems like a trick question to me.
To expand on the numbers a bit more... In scenario (2) the possible combinations are:
(Combo A) G G (0/49 at least one boy TB)
(Combo B) B G (7/49 at least one boy TB)
(Combo C) G B (7/49 at least one boy TB)
(Combo D) B B (13/49 at least one boy TB)
If the man has one boy, then combo A does not apply and the probabilty he has two boys must be 13 / (7 + 7 + 13) == 13/27.