I have two children. One of them is a boy born on a Tuesday. What is the probability they are both boys?

I think it's 1/2, since you already know one is a boy for sure.

First, your reasoning is wrong because you don't know which child - or even if both - fit the description. But trying to explain why that matters makes many people ignore other considerations.

As the famous Martin Gardner observed in 1959 about this question's predecessor, it is ambiguous. I have to know why you choose to tell me this very odd piece of information.

If I walk up to people at random on the street, and ask "Pardon me, but do you happen to have exactly two children, including at least one boy who was born on a Tuesday?" Then the answer is 13/27. There are 14*14=196 possible combinations of Gender+Day, but only 27 of them include a Tuesday Boy. Of those 27 combinations, 13 are two boys.

But if you volunteered the information, I have to assume that you randomly choose one of the one (possible but unlikely) or two (most likely) descriptions that applied to your children. In that case, I have to consider the possibility that you would have said "One of them is a girl born on a Monday" even though her brother was born in Tuesday. So of the 27 combinations with a Tuesday Boy, you would tell me that fact only 14 times. Of those 14, 7 include two boys, and the answer is 1/2.

To determine the probability that both of your children are boys, given that one of them is a boy born on a Tuesday, we need to break down the possibilities and calculate the probability.

Let's assume that there are only four possible combinations of genders and birth days for your two children: BB (both boys), BG (one boy, one girl), GB (one girl, one boy), and GG (both girls).

Out of these four possibilities, we know that one of your children is a boy born on a Tuesday. This eliminates the GG case because both children need to be girls. We are left with three possibilities: BB, BG, and GB.

Now, we need to calculate the probability of each possibility. Let's assume each possibility is equally likely. Therefore, the probability of having a boy born on a Tuesday is 1/7 (there are seven days in a week, so 1/7 represents the probability of a boy being born on a specific day).

Probability of BB (both boys):
Since there are seven possible days for the second child's birth, the probability of having a boy born on any given day is also 1/7.
Therefore, the probability of both children being boys is (1/7) * (1/7) = 1/49.

Probability of BG (one boy, one girl) or GB (one girl, one boy):
In this case, we have two possible gender combinations (BG or GB), and the boy can be born on any of the seven days of the week.
Therefore, the probability of having one boy and one girl is (1/7) * 2/7 = 2/49.

Adding up the two cases where we have at least one boy (BB and BG/GB):
The probability of having at least one boy is 1/49 (BB case) + 2/49 (BG/GB case) = 3/49.

Thus, the probability that both of your children are boys, given that one of them is a boy born on a Tuesday, is 1/3.

It's important to note that this analysis assumes no additional information about the children (such as the birthdays being independent events), and the assumption that each gender has an equal probability of occurring.