Hands Across the Poker Table

          
I have to address this sooner or later.

People play poker. Life is like poker.

There are 52 cards in a standard deck. A poker hand is made up of 5 random cards.  At least, they better be random.

According to Wikipedia, there are just under 2.6 million unique poker hands that you can be dealt. To be exact, there are 2,598,960 different five-card hands.

At a poker table with 8 to 10 players playing a 5-card game, a professional dealer will deal on average about 25 hands an hour. Don’t ask me how I know this. Most dealers would like to deal about 30 hands an hour, but the average will almost always be less than that.

If our poker player plays poker for 7 hours a day, 6 days a week, our player will play 42 hours of poker a week. Each week the player will be dealt 1050 hands. Over the course of a 50 week year (even a workaholic poker player needs a couple of weeks off each year), the player will be dealt a total of 52,500 hands. Our poker player will need 50 years of steady day in day out poker playing to see all 2.6 million unique hands, assuming she is never dealt the same hand twice.

Seeing all 2.6 million unique hands is simply not going to happen during a lifetime of 'bricks and mortar' poker playing. Not even ardent poker professionals have that kind of steadfast time available for constant play. Somebody they know is inevitably going to get married and they will have to miss an extra week or two.

But we now have to make this review a bit more complicated, by adjusting our analysis to account for the current most likely poker situation. We originally postulated a simple 5-card game (as in 5 card draw). Everyone has seen this type of game depicted in Hollywood western movies.  Somebody almost always gets shot.

Most poker players today play Texas Hold ‘Em.  Players rarely get shot. While the ultimate winning hand is still made up of 5 cards, the current game is structured with each player receiving 2 unique (hidden) cards, and 5 community cards dealt face up in the middle of the table.

Each player constructs (mentally) the best 5-card hand she or he can out of the 7 available cards. This makes the total number of unique playing situations considerably greater than the 2.6 million unique hands previously discussed.

Our player can have 1,326 (52 x 51 divided by 2) different 2-card starting hands. If there were only 1 other player at the table, that player could have any of another 1,225 (50 x 49 divided by 2) unique starting hands. There would then be exactly 1,712,304 different 5 card combinations left for the face up community cards. Finally, our poker player’s initial 1,326 different starting hands reduces to 169 different unique hand values. (Eg., for pocket pairs, it does not matter which suits the cards are.)

This leaves our player facing the following total possible relevant combinations of hands to have to evaluate in a two person (heads up) game of Texas Hold ‘Em: 169 x 1,225 x 1,712,304 = 354,489,735,600. That’s 354 billion relevant hand combinations, not 2.6 million unique hands. Rather than 50 years to see all the possible combinations, it will now take our player more than 5 million years to see them all.  The number 354 billion then skyrockets to over 622 quintillion possible hand combinations if there are 9 other opponents rather than just 1 other opponent in the game.

Wowie, wow, wow! That’s a lot of different combinations to have to deal with. There is no chance one person is ever going to see all those different situations. In fact, each player at the table is never going to see but a small, teeny tiny fraction of all the possible combinations of random events that will occur in making up the hands that come across the poker table.

But that’s kind of the way life is, isn't it?


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Update: 11/25/12, follow links to -

The Morning News: Pascal's Wagering
Somewhat Abnormal: Space Elephants!
RandomWalk: God is Possible Argument-I
UA-H: Virtual Labs-Probabilities (see Chap.12, Games of Chance)