hmm... the regular one? I only know one kind. If I had to guess, it involves a lot of calculation. I'm interested in both the calculation and the answer, but I wouldn't know where to start oin my own. I suck at probability and statistics.
Maybe you want the probability that a given game is somehow winnable? I guess you'd have to start by figuring out what sorts of conditions make a win impossible. Almost surely someone has worked on this before, even if they just give monte carlo methods. Maybe try searching for the fancier name of 'klondike'.
In Video Poker, for example Jacks or Better, 9/6, the return to the bettor is said to be 99.54% with "perfect play." The chances of a Royal Flush are every 40,390.55 However, in Double Bonus 10/7 the chances for a Royal Flush, using of course the same deck, are one in 48,048.04.
This very detailed mathematical analysis definitively depends upon computers, and before computers these things could not be figured. First of all, you must discover what is for the bettor perfect play. By simple computation it was often possible to get general figures about what to hold and discard, but to get an exact analysis is another matter and depends upon a computer.
Double Bonus, 10/7, has a supposed payout of 100.1725% with perfect play-- which, of course, assumes truly random dealing. This was certainly not known before computers, because the best strategy for the player was unknown. In fact, in Georgia, for example, it is illegal to have a offer an electronic game that pays back over 100%. Double Bonus was offered because its theoretical payout with perfect play was incorrectly estimated.
This is something like camera lenses, where before computers they would stumble on a formula for a lens, but after computers they really got going and discovered how to make a lot of new camera lenses.
Of course, Video Poker is a paying game, unlike solitaire, and, maybe no one has really put that much effort into discovering the best strategy.
For example, here's one that isn't winnable:
for the first 7 face up cards, you have
2, 2, 2, 2, k, k, k
where under each face up card you have 0,1,2,3,4,5,6 face down cards respectively.
So you have 31 cards left over, and you flip them every three. Then there are 10 cards that you will never be able to use in the rest of the deck if you can't use any of the 21 cards that you will cycle through in a period 2 limit cycle. Just imagine grouping the 31 cards in 10 groups of three, and one extra, you will cycle through, seeing all of the first cards of each group, then you'll see the last card, then you'll cycle through seeing the 3rd card of each group, then the last card, then you will repeat the cycle. if all these 21 cards are the three's, fours, fives, sixes, sevens, and one eight of any suit, then you can't play anything to begin with.
In some cases, you can make many moves, but you will end up on some periodic cycle.