This gives a rough estimate of the probability p that a random game is winnable: p ≈ W/N. The basic idea is to choose a large, yet manageable number N of games and then determine the number W that are winnable.
This is a standard technique in probability and statistics. The answer is to make an estimate using the Monte Carlo method. In the case of solitaire, even the fastest computers would take an impossibly long time to run through all the possible deals and count all the winning hands. For example, the probability of rolling an 8 when tossing two fair six-sided dice is 5/36 since there are 36 possible outcomes and only 5 of them add up to 8. The ratio of these numbers is then the probability. The standard strategy for computing probabilities of certain outcomes is to count the number of possible events and then count the number of outcomes you are interested in. These games aren't all distinct since, for example, we could interchange the two red suits card for card and obtain the same game, but the number of unique games is still very large ( Wikipedia says, without citation, that it's about 7 × 10 15 but that seems too low). A shuffle of the deck determines a game of solitaire. That is a huge number to give you a sense of how huge note that there are about 10 80 atoms in the universe. × 3 × 2 × 1, which is approximately 8 x 10 67. The number of possible shuffles of a standard deck of cards is 52! = 52 × 51 ×.
What's preventing a calculation of the odds? The sheer size of the problem. As it turns out, the answer is unknown, which surprised me a bit since it is such a well-known and popular game. That of course got me thinking about the probability of winning any particular game of solitaire. Under that scenario I won roughly one game in six, with a win percentage of about 17%. With the old fashioned app I might restart a game a couple of times to see if I could win before giving up and moving on.
In the end, I won, after approximately 100 restarts and 4:30:00 in (nonconsecutive) playing time (good thing it was a long weekend). Note the 2:59:39 at the bottom right that's how long I had been trying to win this game at the time. With the old version if I didn't win on the first try I would simply start a new game, but now, since I know the game can be won, I will restart it a few times until I figure out the right sequence of moves to get the win.
At first, I wasn't convinced I would like having every game winnable since it seems like cheating, but as it turns out I rather enjoy it. The app allows games to be restarted so that you can try again if you get stuck. If you make a wrong move at some point, you may lose. That is, every game can be won, although it may not be completely obvious how to do so. A recent update to the solitaire app on my phone installed only winning deals.
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