In the last post, I covered two mathematical formulas useful for calculating advanced odds for poker hands – Texas Hold’em in particular. That post covered the basics. Let’s get in to some more details. (Please re-read the previous post before continuing, as the latter-half of this post is fairly math-dense.)
There are 10 categories of poker hands. They are, from highest value to lowest:
(1) Royal flush = Ace through 10 in the same suit)
(2) Straight flush = Five consecutive cards in the same suit
(3) Four of a kind = Four cards of the same value
(4) Full house = Three of a kind plus a pair
(5) Flush = Five consecutive cards in value, any suit
(6) Straight = Five cards of the same suit, no sequence necessary
(7) Three of a kind = Three cards of the same value, any suit
(8) Two pair = One pair of one card value, and another pair of another value.
(9) One pair = Two of a kind
(10) No pair
These are the poker hands that can be dealt from a single deck of 52 cards. These are just as applicable to Texas Hold’em as other forms of poker. Let’s have a look at some of the total possible unique hands for each category. Recall from the last post that there are 2,598,960 unique 5-card hands possible from a standard deck of 52, where order of cards does not matter.
Based on the definition, there can only be 4 possible royal flush hands because there are only four suits in a deck. These are very rare, obviously. The odds of being dealt 5 straight cards to form a flush are 4/2,598,960 =~ 1/650,000. [Note: these apply to the scenario where you deal 5 straight cards from a deck of 52. Obviously, in a game of poker, there are several players. So the odds of a royal flush are even rarer in a real game.]
There are 9 different straight flushes for each suit: K=>9, Q=>8, …, 5=>A. Since there are 4 suits, there are a total of 9×4 = 36 possible unique straigh flush hands, excluding the royal flush. In Texas Hold’em, your chances of scoring a straight flush diminsh greatly, if you haven’t flopped a straight flush draw before 4th and 5th street. The odds are 36/2,598,960 =~ 1/72,000. On the other hand, if you include the royal flush, then the odds are (10×4)/2,598,960 =~ 1/65,000. [Some poker guides use this figure for the odds of a straight flush.]
Four of a Kind
There are only 13 different card values in a deck. So there are only 13 possible four-card hands. However, there has to be a fifth card to make up a full poker hand. We’ve used up 4, and have 48 left. So, the total number of unique four-of-a-kind hands is 13×48 = 624. This is because there are only 48 cards left after the four of a kind are dealt. Remember, order doesn’t matter. So the odds of a four-of-a-kind hand are 624/2,598,960 =~ 1/4000.
Three of a Kind – cbetcasino.fr
There are 13 possible values in 4 suits. The first three cards in this case must be all the same. For a given value, say Q, there are B(4,3) = 4! /3! (4-3!) = (4×3×2×1) /(3×2×1)(1!) = 4 unique triplets. Times 13 values = 52 ways to pick the first three cards to form only trips.