Smash Vegas

All poker variants share the same system of hand rankings.  But how are they decided?

Quite simply it’s down to the likelyhood of the hand occuring, with rarer hands worth more.  We can calculate the probability of being dealt a hand by looking at the number of ways five cards can make that hand.

There are 311,875,200 ways (permutations) of being dealt five cards from a 52 card deck.  However since a poker hand has the same value whatever order the cards are arranged in we are interested in the number of unique ways (combinations).  To find this we divide by the number of ways a five card hand can be arranged, which is 5! = 5*4*3*2*1 = 120.

Thus the number of distinct hand combinations is 311,875,200 / 120 = 2,598,960.

Since a royal flush is just a particular (the best) straight flush we start by enumerating the straight flushes.

A straight flush is a hand of five cards in sequence, all of the same suit.  When comparing straight flushes, the hand with the highest card wins.  Suits have no relative value and are all ranked equally.  Obviously there are 4 royal flushes, and 36 other possible straight flushes.  So the probability of being dealt a straight flush is 40 / 2,598,960 ≈ 0.0015%.

Four of a kind (or quads) is fairly self explanatory - four cards of the same rank plus one more unmatched card.  Comparing quads the highest matched set wins.  If two players have the same four of a kind (i.e. in a community card game) then the player with the highest kicker (the unmatched card) wins.  There are 624 different ways to make a hand with four of a kind, hence the probability of being dealt one is 624 / 2,598,960 ≈ 0.024%.

A full house is a hand containing three cards sharing one rank, and two cards sharing another rank.  Between two full houses the hand with the higher ranking set of three wins.  If the set of three mayching cards is the same then the hand with the higher pair wins.  There are 3,744 possible full houses and so the probability of being dealt one is 3,744 / 2,598,960 ≈ 0.14%.

A flush is five cards all of the same suit, but not in sequence (otherwise it would be a straight flush).  As the suit has no value a flush with a higher ranking high card is considered superior.  There are 5,148 possible hands with all five cards sharing the same suit but 40 of these are straight flushes, so the probability of being dealt a flush is 5,108 / 2,598,960 ≈ 0.20%.

A straight is a hand of five cards in sequential order but of more than one suit (otherwise it would be a straight flush).  Note that as with straight flushes an ace may be high or low but not both (i.e. a straight can not be “around the deck” and include both a king and a 2).  Straights are compared by looking at their highest card - remember that if the ace is low it counts as a one!  There are 10,240 hands with all five cards in sequence but 40 of these are straight flushes so the probability of being dealt a straight is 10,200 / 2,598,960 ≈ 0.39%.

Three of a kind (also known as trips or a set) is a hand containing three cards of the same rank plus two unmatched.  As usual a higher ranked three of a kind beats a lower valued one, and if the rank is the same then kickers decide.  There are 54,912 hands containing three cards of the same rank which are not full houses.  Thus the probability of a three of a kind is 54,912 / 2,598,960 ≈ 2.1%.

Two pair is a hand containing two pairs of cards of the same rank, plus one unmatched card.  If two players both have two pair then the win is decided first by the high pair, then the low pair, then the kicker.  There are 123,552 two pair hands that are not full houses.  The probability of being dealt one is 123,552 / 2,598,960 ≈ 4.75%.

One pair is a poker hand containing a single pair of cards of the same rank, with three unmatched cards.  If two players both have one pair then the higher pair wins.  If they both have the same pair then it is decided by the kickers.  There are 1,098,240 different hands with a single pair.  Thus the probability of being dealt one of them is 1,098,240 / 2,598,960 ≈ 42.26%.

A high card hand is one with none of the above properties.  It does not have any cards of the same rank, they are not all in sequence and they are of more than one suit.  When comparing high card hands the one with the largest high card wins.  Since we have eliminated the other options we know there are 1,302,540 different ways to be dealt a high card hand and so the probability of being dealt one of them is 1,302,540 / 2,598,960 ≈ 50.12%.

Hope that clears up why the hand rankings are the way they are and how to compare hands of the same type.