The issue here is with assumptions - you have made a different set of assumptions from the author, and hence are getting a different result.
A lot of people here are having similar issues, by misreading exactly what the initial proposition means.
Your reasoning above relies on the 'likeliness' of a man giving you the information, which is something that is not meant to be a part of the problem. Although it is phrased as a man 'telling' you something, that statement is really a metaphor for 'you determine the following piece of information, 100% truthfully'.
In particular, your explanation assigns agency to the man - that if he does in fact have a son, he may or may not choose to reveal the truth 'I have a son'. However it makes no allowance for the man lying - so you are assuming if he answers it will be truthfully, but you are allowing him the lie of omission.
Whilst there is nothing specific that rules out your interpretation, it is not what is intended. Read it instead as:
----
There exists a man, A.
A has exactly two children.
The statement 'A has at least one Son, B' is true
What is the chance that the statement 'The Non-B child of A, is a son' is true?
I agree that that is what the author intended to communicate. But I actually think that's a bigger stretch than my own interpretation - it changes how the information was determined, which has a definite impact on the outcome.
The reason why I posted was to suggest that the author should have worded it the second way, i.e.
A man has two children, and one is a son born on a Tuesday. What is the probability that the other child is also a son?
Which leaves no doubt. I guess I didn't really make that clear enough with my original post.
I'd also observe that carefully read, this article is really about how important assumptions are, and not about the problem per se. The Peter Winkler quote is key.
A lot of people here are having similar issues, by misreading exactly what the initial proposition means.
Your reasoning above relies on the 'likeliness' of a man giving you the information, which is something that is not meant to be a part of the problem. Although it is phrased as a man 'telling' you something, that statement is really a metaphor for 'you determine the following piece of information, 100% truthfully'.
In particular, your explanation assigns agency to the man - that if he does in fact have a son, he may or may not choose to reveal the truth 'I have a son'. However it makes no allowance for the man lying - so you are assuming if he answers it will be truthfully, but you are allowing him the lie of omission.
Whilst there is nothing specific that rules out your interpretation, it is not what is intended. Read it instead as:
----
There exists a man, A.
A has exactly two children.
The statement 'A has at least one Son, B' is true
What is the chance that the statement 'The Non-B child of A, is a son' is true?
----