In the Vice (HBO) clip AI Poker Bots Are Beating The World's Best Players, it is claimed (at 2:00), that Texas Hold'em has "more possible hands than atoms in the universe".

..this competition featured a complex style of the game called Heads Up No Limit Texas Hold'em, with unlimited bet sizes and more possible hands than atoms in the universe.

Due to the insane size of the universe and the tininess of atoms, I got skeptical of the validity of the claim.

The presenter does not specify what she considers "a hand". For now, I assume she means a unique combination of hole cards and community cards.

We are talking about Heads Up, so there are only two players, each of them gets two hole cards. The order of the hole cards is irrelevant, so the deals of hole cards "Ac Kd" and "Kd Ac" should be considered identical, as they play exactly the same way.

The order of the cards of the flop is also irrelevant, in the same way (in some cases one can get live reads based on order if the flop is dealt in a certain way, but this clip deals with computerised games.)

Then there is the turn and river. The order of the flop, turn and river obviously matters.

The game uses a deck of 52 cards, and all in all, a total of 9 cards are dealt, two to each player and five community cards.

I guess one could also interpret the claim to include player actions as a part of a "hand", so each call, check, bet, raise or fold would be a part of the hand. If this is the intended meaning then I guess it becomes sort of a no-brainer. Since it is no-limit, you can make any arbitrary combination of stack sizes, blinds, and bet/raise sizes, and thus create and infinite combination of "hands" in that way.

Related: Are there more 40-moves chess games than atoms in the universe?

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    That number should be bounded by the number of permutations of the whole deck which is 52!. That number is 80658175170943878571660636856403766975289505440883277824000000000000 which is about 10^67. The number of atoms in the universe is about 10^80 (although estimates vary a lot)so not even close – Giacomo Alzetta Jan 16 at 12:57
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    Yep, this is easily tested by calculating the number of unique hand combinations (a simple math problem, once you know the rule for dealing) and comparing to scholarly estimates of the atoms in the universe. – Daniel R Hicks Jan 16 at 13:05
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    @GiacomoAlzetta Seems like an answer, then, not a comment? – Revetahw Jan 16 at 13:22
  • If they meant a poker hand as an entity which includes player actions and bet sizing (not to mention players themselves), then THAT number is in fact infinite. – Alexandru Clonțea Jan 16 at 13:43
  • @AlexandruClonțea Yes, as it says in OP: "Since it is no-limit, you can make any arbitrary combination of stack sizes, blinds, and bet/raise sizes, and thus create and infinite combination of "hands" in that way." – Revetahw Jan 16 at 13:46

According to Superhuman AI for heads-up no-limit poker: Libratus beats top professionals Science 26 Jan 2018: Vol. 359, Issue 6374, pp. 418-424, Head-Up No Limit (HUNL) Texas Hold'Em

has 10^161 decision points (24)

The footnote explains:

The version of HUNL that we refer to, which is used in the Annual Computer Poker Competition, allows bets in increments of $1, with each player having $20,000 at the beginning of a hand"] has 10^161 decision points [reference 24: Measuring the size of large no-limit poker games (Technical Report, Univ. of Alberta Libraries, 2013)]

This is much larger than the estimate of 7.1×1079 atoms in the universe from Physics.SE.

If you are just talking about the deal alone, and not decision points, it can be calculated with high-school maths:

(52*51)/2 possibilities for the hand of player 1.

(50*49)/2 possibilities for the hand of player 2.

2 possibilities for who is dealer

(48*47*46)/6 possibilities for the flop

45 possibilities for the turn

44 possibilities for the river

(2*52*51*50*49*48*47*46*45*44)/24 = 111255240096000 = 1.1 x 1014.

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    Which Wolfram Alpha informs us is less than 6 times the number of red blood cells in your body. So.... no. ;-) – DevSolar Jan 16 at 13:54
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    @GEdgar The OP also specified that including betting actions in the definition of "a hand" would trivially make the statement true, so it didn't need to be addressed by answers. – Kamil Drakari Jan 16 at 15:31
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    @Revetahw: That interpretation would be open to criticism that a difference in $1 hardly makes for "a different hand". And if you categorize bets, to e.g. "about a quarter of the opponent's chips", "about half the opponent's chips" etc., you again end up with "hilariously less variations than claimed". – DevSolar Jan 16 at 16:12
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    @DevSolar Not only that, but the quote from the OP distinguishes between the two - "unlimited bet sizes AND more possible hands...". Which, by common English, indicates that independent of bet sizes, there are more possible hands than atoms, which means that whatever criteria they are considering for a hand, it does not include bet size. – cpcodes Jan 16 at 17:11
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    @Oddthinking For your first comment, it depends if you consider the players themselves (with generally two different stacks) indistinguishable. For the second comment, I agree the games are identical, but I still consider them to be different hands. For both comments, I'm trying to err on the side of a big number, to give the benefit of doubt to the claim, and still it is a factor of 10^65 too low. – DavePhD Jan 17 at 13:46

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