Number of Possible Limit Hold'em Games

[JEM7VSBL]

I was just wondering the other day if anyone has been able to calculate the total number of possible outcomes for a limit hold'em game, assuming a 10-handed game with a normal blind and capped betting structure. there is definitely a finite number of possibilities, but is the number too large for a computer to calculate?

we can start with the 52! (factorial) to get the base number of card orders in the deck. then each player has the option preflop to fold, call, or raise. based on each action, 9 other players have 3 options, and so on, for each player and each action. same goes for the three postflop betting rounds with options to check, call, bet, raise, fold.

has anyone developed an algorithm for calculating this, much less even cared how big this number could be? i'm just curious.

[donkeyradish]

Its such a big number we'll never see the same game being played twice anyway

[Daniture]

what are you talking about. The hand that just happened is equally likely to happen again next hand as any other outcome. and i would not be suprised if it has already happened somewhere.

[LetYouDown]

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what are you talking about. The hand that just happened is equally likely to happen again next hand as any other outcome. and i would not be suprised if it has already happened somewhere.

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I imagine he's talking about the fact that there are 7.4074E+39 ways to order 52 cards in 25 spots. Let alone factoring in the bet/raise/call/fold actions.

[murellus]

is the game capped when betting is heads up?

[AaronBrown]

It's not quite 52! for the cards. Only 25 cards are dealt, and order doesn't matter within a hand or within the flop. So it's 52!/[27!*2^10*6] = 1.2*10^36 while 52! is 8.1*10^67. And not all the cards matter for every hand, if it's everyone but the BB folds preflop then the board doesn't matter.

It would not be hard to count all possible betting outcomes by computer program, if you restrict each round to three raises in all cases. If everyone stays in to showdown and every betting round is capped then there are one trillion ways to distribute the raises. The average player makes 11 calls, any one of which we could replace by a fold, which makes 11^10 = 26 billion ways. Not all of those work, for example if everyone folds preflop, there's no one to raise post flop. So, 26 billion trillion, or 2.6*10^22 is an overestimate of the number of possible betting scenarios, I'd knock off a couple zeros to eliminate the cases in which someone folds before they raise, to get something like 10^20. Anyway, it's much less than the number of card combinations.