Let’s take a specific example, say 5 players for Texas Hold’em, and review the rules so that it’s easierto follow the formula. After the blinds (small and big) are bet, each player is dealt 2 cards, starting to the dealer’s left. Then the next card on the top of the deck is “burned”, followed by the turn of the first three cards of the flop. The next four cards are: burned, turned, burned, turned.

 

So the set of m-values is {2, 2, 2, 2, 2, 1, 3, 1, 1, 1, 1}, even though there are only 5 players. For each additional player, you have to add another “2” to the set. Now we can make a relatively simple calculation.

 

How many many possible ways a deck of 52 cards be dealt out to 5 players, according to the rules of Texas Hold’em, and where the order of cards in a player’s hand does not matter? The order of cards between players also does  not matter, at least not for the initial calculations we’ll consider. We’re treating the separate burn and turn cards as if they are players, too.

 

The answer is M(52, 2, 2, 2, 2, 2, 1, 3, 1, 1, 1, 1) = 52! /2! 2! 2! 2! 2! 1! 3! 1! 1! 1! 1! = 52! /(2^5) 3! = a very large number, with 66 digits. A lot larger than the number of 5-card hands we saw previously (311 million-plus

 

You might think that the “1” values are not doing anything, since 1! = 1, but consider this. If you grouped the burn and flop cards (including 4th and 5th street) into one group, for a total of 1+3+1+1+1+1 = 8, you would eliminate the combinations where a certain card ends up in a turn pile instead of in the community cards. You’d also eliminate combinations where one card shows in the flop rather than in, say, 4th street, or vice versa. We want the extra-group order to be important, because where a card appears in the flop, 4th or 5th streets matters to the betting in Texas Hold’em. More cbetcasino.fr

 

So if we grouped those cards, we’d be eliminating too many combinations that we want to count. Thus the formula as it stands above ensures that we count all unique combinations, but where order of cards within a group doesn’t matter.

 

I know that’s not much of an example, but it’s a start. I’ll get in to some detailed examples in the next part of this series (which may be two posts ahead). If you can handle the math, probably 4th-year university Calculus or beyond, there is an extremely advanced math paper (PDF) that discusses a poker player’s “risk of ruin”. (It’s partially based on the size of the bank roll you start with, and your win rate.)

 

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