By "solitaire", let us mean Klondike solitaire of the form "Draw 3 cards, Re-Deal infinite".
What is the probability that a solitaire game be winnable? Or equivalently, what is the number of solvable games?
When I came up with the question, it seemed a pretty reasonable thing to ask, and I thought "surely it must have been answered".
I have no probability formation (save for an introductory undergraduate-level course), but anyway I started thinking on how could the problem be tackled.
Immediately my interest shifted from the answer to the above question, to the methods involved in answering it. I couldn't even begin to figure out how would one go solving this problem!
How does one even begin to find the number of solvable games?
In the same wikipedia link, it is stated that
For a "standard" game of Klondike (of the form: Draw 3, Re-Deal Infinite, Win 52) the number of solvable games (assuming all cards are known) is between 82-91.5%. The number of unplayable games is 0.25% and the number of games that cannot be won is between 8.5-18%.
The reference for the thresholds is this paper by Ronald Bjarnason, Prasad Tadepalli and Alan Fern.
It came as a surprise to me that the answer is not really known, and that there are only estimates. I tried reading the paper, but I'm too far away from those lines of thinking to understand what they're talking about. There seems to be some programming going around, but what is the big idea behind their approach to the question?
I would like to end this question with a couple of lines from the paper (emphasis by me):
Klondike Solitaire has become an almost ubiquitous computer application, available to hundreds of millions of users worldwide on all major operating systems, yet theoreticians have struggled with this game, referring to the inability to calculate the odds of winning a randomly dealt game as “one of the embarrassments of applied mathematics” (Yan et al., 2005).