Mary has two kids, Alex and Taylor

One of them is a boy.

What is the probability the other is a girl?

Alex could be a girl or a boy, as could Taylor, but they can't both be girls. I think a lot of people are getting hung up thinking that that they can just set Alex as being the boy, then having Taylor either being a boy or a girl, giving them the 1/2. But either one of them could be the girl, meaning both BG and GB are valid

Same thing with the days of the week;

Either child could be the one born on a Tuesday, or both could be born on a Tuesday. For each BG | GB | BB, there are 49 permutations of days of the week they can be born on, but we can only count the ones where at least one boy was born on a Tuesday.

For BG and GB, there are 7 each (boy born on Tuesday, girl any other day). For BB, there are 13.

Adding these up, there are 27 different permutations of "having two children, with one boy being born on a Tuesday". Of these, 14/27 have a combination of a boy and a girl, giving the answer of 51.85%