Not to mention that the first move itself is a guess. I'm not sure about all variants, but certainly on the classic Windows minesweeper there were times that my first click was a bomb.

That was definitely not possible on Windows XP version of Minesweeper, they would regenerate the board if you clicked on a bomb in the background and used that one instead.

Oh, really? I guess I was just misremembering things. Thanks for correcting me here! (not sarcastic, genuinely happy to be corrected when I remember things wrong)

It was definitely not like that on Win98.

Definitely not like what the parent said, or what I said? Maybe I'm remembering that rather than XP

You would think that if you clicked a bomb first, the app would regenerate the board until it made one where the place you clicked was not a bomb.

Some games do this, some don't some only actually generate the field once you click the first box.

The behaviour towards the user is the same, though: The first click never results in a bomb. Everything else is an implementation detail, although not placing bombs until after the first click is probably the easiest approach to ensure it.

I was thinking that past a certain threshold of moves the probability of clicking a mine in most strategies will drop to zero, the question I had was if there were a specific strategy where that probability would never increase past 0.

From all the commentary however, it's pretty clear that the answer is no.

Consider the following configuration (where --- is the wall and M == known mine):

  +------
  |     1
  |     2
  | 1 2 M

If you know there are three mines here, including the marked one, this is either

  +------
  | M   1
  |   M 2
  | 1 2 M

or

  +------
  |   M 1
  | M   2
  | 1 2 M

There's no way to ever get more information to resolve the ambiguity without guessing, and the mine probability in every square is 50%. This (or simple variants) is the most common late-game failure case. Similar symmetries can also occur on edges, and (though more rarely) even in the middle of the field.

If you assume that you have knowledge of the 5 outside squares of a 3x3 corner (and any neighbors outside of that 3x3 corner) and that you know how many mines are left, you won't be able to fill in the rest without guessing 4.14% of the time on an Expert board. The most common failures are either three three-mine cases like the one illustrated here (0.440% likelihood each) or one of 16 four-mine cases (0.115% likelihood each). Given that there are four corners on the board, that sends your failure rate to at least 15.572% (actually higher, because you don't always know how many mines are left in one corner without clearing the other three).

If you want to fail fast, figure out if you have symmetry in the corners first.