Except for game number 11982 (in the older version of Free Cell, which gives you 32000 starting positions). AFAIK, no one has ever beaten that one.
Returning to Klondike, the odds of winning, when you turn over three cards, and can go through the hand as many times as you like, are often given as 1 out of 34. In fact, the odds are 1 out of 17.
How do I know? Back in the dark ages of computing, I programmed a solitaire game (in C - not C++, just plain C), as the infernal machine (an AT&T 3B5 minicomputer) had been delivered with no games (we’re talking old Unix, here, System V, Release 2). I quickly realized that I had, of necessity, built into the game a little subroutine to determine whether a move was legal or not. It was an easy matter to alter the program to play by itself, simply by trying every possible move, and making any move that was legal.
Obviously, I built in enough logic so that it wouldn’t move a king from one empty space to another, or otherwise end up in an endless loop. However, I didn’t go so far as to let it move partial columns, or to “retrieve” cards once they had been put on the “home” piles.
I turned the sucker loose overnight, and it played thousands upon thousands of games, keeping track of its winning percentage. To within a couple of decimal places, it won 1 game out of 17. Of course, a human player, making some informed choices, and able to move partial columns, would be able to beat those odds.
Why are the odds usually given as 1 in 34? I can only speculate, but I believe that someone once worked out the probabilities, and made a simple error somewhere along the line, leading to a result that was off by a factor of 2. Since then, no one has ever bothered to figure it out again, and the wrong figure has simply been repeated so often that it’s accepted as correct.