In all likelihood, Gary Foshee has two children either of different genders, or born on different days of the week, or both. If he does, he had to choose between two problems of the form “I have two children. One is a (gender) who was born on a (weekday). What is the probability I have two (gender)s?”, where the words in parentheses with appropriate values. Now, it is true that 13/27 of all families that any such statement S applies to will have two of the named gender child, but existence is not the same thing as probability. To get probability you have to also factor in the probability Gary Foshee will choose one the particular statement he did.

So, for one case in 196, Gary Foshee has two Tuesday Boys and makes statement S with probability 1. In 12 cases, he as two boys but only one was born on a Tuesday, and he makes statement S with probability P. And in 14, his Tuesday Boy has a sister, and he also makes statement S with probability P. The resulting probability of two boys is:

P(two boys) = (1+12P)/*(1+12P+14P) = (1+12P)/(1+26P)

It is now easy to see that the answer is 13/27 if, and only if, P=1. That is, if Gary Foshee somehow preferred to tell you about a Tuesday Boy over any other combination. Since nothing in his statement suggests that, it is wrong to assume it. The more reasonable value is P=1/2, in which case the answer is 1/2. This same analysis can also be applied to the original problem, without Tuesday, and the answer is 1/3 if and only if Gary Foshee prefers to tell you about boys over girls. Again, the more reasonable assumption is P=1/2, in which case the answer is also 1/2.

The way you worded the problem, one could (and many do) argue that P=1, because it is a statement about existence. As Gary Foshee actually worded it, it was a choice he made and P must be 1/2. Ironically, this change in the answer based on how you learned the fact was first pointed out by the man Gary Foshee was supposedly honoring, Martin Gardner, when he retracted his original answer of 1/3 in October 1959.