You're making the exact mistake the author is cautioning against, which is assuming the day doesn't matter. Write out all the possibilities (see my other comment in this thread), eliminate the dupe, and you get 13/27.
But the fact that the father voluntarily offered up the information changes the probability distribution. We can assume he's selecting one of his children at random, and revealing their birthday and gender.
If only one of his children is a male/Tuesday, there's a 50% chance he'll say male/Tuesday.
If both are, there's a 100% chance.
So I'm not counting the possibility twice; I'm saying that given that the father reveals male/Tuesday, it's disproportionately likely that's as a result of having two male children born on a Tuesday compared to any other possibility.
This is the entire point of the article, IMO: we have to make some assumption about how we selected this guy to talk to, and how he chose what to tell us. It's not pinned down by the statement of the problem, and what you might consider a natural assumption is not necessarily what other people might assume.
Which is why these problems tend to suck...
But even given other assumptions as to why the father selects the child he does - sort by date, males first etc, the author's answer of 13/27 is still almost certainly wrong - he should have just taken the father out of the equation completely.
Think of it like this. What would the father say if he did in fact have two boys who were both born on Tuesday. Would he really say, "I have two children and one is a son born on a Tuesday"? Wouldn't he instead say, "I have two children and both are sons born on a Tuesday."?
I mean he could say it the first way, but such a comment would be borderline misleading. To say you have one son born on Tuesday when in fact you have two is technically correct, but I think the problem assumes that the man is speaking somewhat plainly.
So I do agree with you that communication intent is ambiguous, but I agree with some others that this is sort of the whole point of the problem, and it's not always immediately obvious that statistical information is hiding in seemingly irrelevant data.
N = Total number of ways to have 2 children over 7 days = 14^2 = 196
B = Two sons born on Tuesday.
O = Exactly one son born on Tuesday.
A = At least one son born on Tuesday.
T = Two sons.
S = The statement.
P(B) = 1/N = 1/196 = 0.005
P(O) = 26/N = 26/196 = 0.133
P(A) = P(B) + P(O) = 27/N = 27/196 = 0.138
P(S|O) = .5
P(S|B) = 1
P(S) = P(S|B)P(B) + P(S|O)P(O) = 1.0 * 0.005 + 0.5 * 0.133 = 0.0715
P(B|S) = 1/27 = 0.037
P(O|S) = 26/27 = 0.963
P(A|S) = 27/27 = 1.0
P(T|S) = 13/27 = .481
P(S|A) = P(S|B)P(B|A) + P(S|O)P(O|A) = 1.0 * 0.037 + 0.5 * 0.963 = 0.519
The best analogous problem is the German Tank Problem:
http://en.wikipedia.org/wiki/German_tank_problem
If we have destroyed a single German tank with a serial number 100, we can at least to begin to make an estimate on the size of the German force, by basically asking the question:
"If they have 200 tanks, what was the chance one we randomly killed was this serial number? 500? 1000?"
And then combining n=100->infinity to form a probability distribution. You can then say that there is an x% chance that Germany has 500 tanks, and a y% chance that Germany has 10,000 tanks.
However - if instead, we asked 'does there exist a German tank with a serial number 100', and the answer is yes, this does NOT tell us anything past the fact that their tanks are >= 100 in number.
We have the exact same information, but how it was determined changes the outcome drastically.
Does that make sense?
I already showed the probability that we would receive the message the way we did, assuming we're sampling fathers with two children, and its pretty low. If we drop the sampling assumption, it would go even lower. But that's irrelevant to the actual question, because we've already won that lottery. I've also shown the probability that we would get the statement we got given that the father had at least one son born on a Tuesday, but again, we already won that lottery.
If you still insist, can you please stop talking in hand-wavy fake math and show some actual concrete numbers? To start, if each of the 27 possibilities are not equally likely, what are the actual probabilities and why?