Effect of number of players in the probability distribution of poker hands

Question

We have this official rank of hands.

I have a theory (which I do not know how to prove) that the number of players in a texas-hold-em table will affect the probability distribution of the hands.

That is, depending on how many players are on the table, the chances of being dealt a certain hand will vary. In the extreme case that the probabilities vary a lot, this would imply that, depending on the number of players, we have the interesting situation that the game is being played with the established rank of hands, but the actual probabilities are not the ones implied by that rank.

Please note that I am not talking about odds, implied-odds, pot-odds or anything related to that. I am only talking about the probabilities of the different possible hands.

Let's keep the assumptions simple: we have N players, all playing until river. Here are my two simple questions:

1. Does the number of players N affect the probability distribution of hands?
2. In case 1. is affirmative: is there a number of players that will cause a "reversal" in the probability distribution (as compared to the official rank of hands). How many players, and what hands get "reverted"?

And to further clarify my questions. If the official rank is

Straight Flush > Quads > Full House > Flush > Straight > Set > Two Pair > Pair > High Card

Is there a number of players N where the actual probability is:

.... > Straight > Flush > ...

This is just an example, any other "reversal" would be interesting. Actually I am interested in evidence that the number of players has an effect (however small) in the probability of the hands, even if no actual "reversal" ever happens.

Answer 1

This can not be true... A flush is always better than a straight, no matter how many players there are at the table.

Now, if you're talking about a higher probability of a player to have a flush instead of a straight... this depends on a lot of factors... and by a lot I mean a lot:

• players are usually tighter when there are 8 players at the table vs. 3 players
• the player's skill is also a factor
• the structure of the game also has a big influence on what cards are being played. You may play K 9 suited if you're at a final table, but fold pocket Tens (a much better hand) if the bubble is about to burst.
• and much more

All these factors influence what hands are folded and which are not. Which means that the probability of a player having a flush or a straight is greatly changed.

Now, if everyone plays their hand and goes to the river, like you said, then the probabilities stay the same. There are 4 cards of the same rank and 13 cards of the same color. If the dealer is shuffling properly, then there's the same probability for each player to get J♣ or K⋄ or 3♣ etc.

Let's say there are 3 players at the table. If you give them 3 random cards (1 to each), then it means that there is:

• a 1/4 chance for each one that they'll get a club
• a 1/4 chance for each one that they'll get a diamond
• a 1/4 chance for each one that they'll get a heart
• a 1/4 chance for each one that they'll get a spade.

25 % FOR EACH SUIT, right ?

This means that, on average, there will be X number of hearts left in the deck, X number of clubs, X number of spades and X number of diamonds. I don't know how much X is and it doesn't matter. All that matters is that X is the same for each suit.

Now, let's do it for 8 players. If you give 1 random card to each one, there will be:

• a 1/4 chance for each one that they'll get a club
• a 1/4 chance for each one that they'll get a diamond
• a 1/4 chance for each one that they'll get a heart
• a 1/4 chance for each one that they'll get a spade.

It's the same 25 %. In this case, on average, there will be Y number of hearts left in the deck, Y number of clubs, Y number of spades and Y number of diamonds.

This means that, just as in the case of 3 players, Y is the same for every suit.

Now here's the key point: The next time you want to give to each of your 3 OR 8 players a card, the probability for each of them of getting a club, diamond, heart or spade is still 1/4, EVEN THOUGH there may be more or less cards in the deck.

This also applies to the community cards. And to the different ranks of cards as well (not just the suits). Which means that there's the same chance of someone making a straight or a flush when there 3 players at the table or 8 players.

So, bottom line:

• if everyone plays every hand and goes to the river, the probabilities are the same
• if not, then they are greatly influenced by a huge number of (sometimes subtle) factors, which means probabilities are kind of useless in this situation....

Answer 2

No matter how many players are at the table, dealt hands are completely random.

After hands are dealt, some players decide to fold their weaker hands so as not to lose money. This means that stronger hands make it to the river.

The more people who are playing, the more likelyhood that someone has a very strong hand, which means that even middling hands will fold.

If you were playing 50-handed poker (ignoring the fact that there aren't enough cards to go around), the only cards worth playing would be connected broadways, ace-high flush draws, and pocket pairs above ~8 or 9. BECAUSE: a) connected broadways can make the nuts if there is no flush or pair on the board, b) ace-high flushes will make the nuts if the board isn't paired, and c) sets and full houses will often make the best hand on wildly disconnected boards.

Answer 3

The issue is what is the probability distribution of the hands that go to the river.

If it is ten-handed, there are ten hands, and the probability distribution (ex ante) of hands is the same from one round to another.

Whether people play "tight" or "loose" determines how many hands go to the river, rather than being folded earlier. In this regard, the number of people playing will determine the probability distribution of hands that go all the way to the end.