Poker Hand Probabilities

The Game of Poker

Poker is played with 52 cards. The cards come in 4 suites: clubs, diamonds, hearts, and spades. Each suit has 13 cards: 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack (J), Queen (Q), King (K), and Ace (A). The card deck is shuffled and each player is dealt with 5 cards. Sometimes players may change several cards in the hope of obtaining a better combination, or a "hand". The game then continues with players either raising stakes or getting out of the game and once the cards of each player are revealed the higher hand wins. There are many aspects of the game, but we will only look at probabilistic properties of poker hands.

Poker Hands

There are 8 commonly used poker hands:

Hand Description Example
Straight Flush 5 cards in sequence and of the same suit. 3,4,5,6,7 of diamonds
Four of a Kind 4 cards of the same rank. 10,10,10,10, and J
Full House 3 cards of the same rank and pair of cards of another rank. 3,3,3, and Q,Q
Flush Five cards of the same suit, but not in sequence. 2,3,10,Q,A of diamonds
Straight Five cards in sequence, but not of the same suit. 8,9,10,J of spades, and Q of hearts
Three of a Kind Three cards with the same value. J,J,J, and Q,A
Two Pairs 2 separate pairs of cards. 5,5, and 10,10, and Q
One Pair One pair of cards of the same rank. 3,7,J, and K,K

Hand	Description	Example
Straight Flush	5 cards in sequence and of the same suit.	3,4,5,6,7 of diamonds
Four of a Kind	4 cards of the same rank.	10,10,10,10, and J
Full House	3 cards of the same rank and pair of cards of another rank.	3,3,3, and Q,Q
Flush	Five cards of the same suit, but not in sequence.	2,3,10,Q,A of diamonds
Straight	Five cards in sequence, but not of the same suit.	8,9,10,J of spades, and Q of hearts
Three of a Kind	Three cards with the same value.	J,J,J, and Q,A
Two Pairs	2 separate pairs of cards.	5,5, and 10,10, and Q
One Pair	One pair of cards of the same rank.	3,7,J, and K,K

Probabilies of Poker Hands

There are

	 52       52!        52*51*50*49*48
	C   = ----------- = ---------------- = 2,598,960
	 5     5!*(52-5)!    5*4*3*2*1

ways to deal 5 cards from a 52 card deck. If there is no cheating and the deck is properly shuffled before dealing, then we may assume that all of these 2,598,960 combinations are equally likely to occur. Therefore, to determine the probabilities of a poker hand we need to find the number of combinations that result in this hand and then divide it by the total number of combinations, 2,598,960. The explanations follow the table.

Hand Frequency Probability "Chances"
Straight Flush 36 0.00001385 1 out of 72,193
Four of a Kind 624 0.00024010 1 out of 4,175
Full House 3744 0.00144058 1 out of 694
Flush 5112 0.00196694 1 out of 508
Straight 9180 0.00353218 1 out of 283
Three of a Kind 54912 0.02112845 1 out of 47
Two Pairs 123552 0.04753902 1 out of 21
One Pair 1098240 0.42256903 1 out of 2.37
or 5 out of 12

Hand	Frequency	Probability	"Chances"
Straight Flush	36	0.00001385	1 out of 72,193
Four of a Kind	624	0.00024010	1 out of 4,175
Full House	3744	0.00144058	1 out of 694
Flush	5112	0.00196694	1 out of 508
Straight	9180	0.00353218	1 out of 283
Three of a Kind	54912	0.02112845	1 out of 47
Two Pairs	123552	0.04753902	1 out of 21
One Pair	1098240	0.42256903	1 out of 2.37 or 5 out of 12

Straight Flush

We have 4 choices for suits and inside each suite we have 9 choices, because any sequence of 5 cards is of the form 2,3,4,5,6 or 3,4,5,6,7 or ... or 10,J,Q,K,A. Therefore, there are 4*9=36 combinations and the probability of having a straight flush is

	    36
	----------- = 0.00001385
	 2,598,960

The reciprocal of this number is 2,598,960/36 = 72,193.

Four of a Kind

There are 13 ways to choose the rank of the 4 of a kind cards (they can be all 2's, or 3's, or 4's, and so on). Once we choose the rank for the 4 of a kind cards, the other (5th) card can be any one of the 48 cards left. So, we have 13*48 = 624 ways (simple events) to have a four of a kind poker hand. The probability is then

            624	
	----------- = 0.00024010
	 2,598,960

The reciprocal of this number is 2,598,960/624 = 4,165.

Full House

There are 13 choices for the rank of the 3 of a kind cards and there are "4-choose-3"=4 ways to have different suites of these 3 cards. For example, we can have a full house with 2,2,2,?,? or 3,3,3,?,? or ... or A,A,A,?,? and 2,2,2 can be either of spades,hearts,diamonds or spades,hearts,clubs or spades,diamonds,clubs or hearts,diamonds,clubs. So, there are 13*4=52 ways to get 3 cards of the same rank (in different suites). Once we picked 3 cards of the same rank, the other two cards must form a pair of different rank. There are 12 choices for the rank of the pair and "4-choose-2"=6 choices for the suits in the pair (spades,diamonds or spades,hearts or spades,clubs or diamonds,hearts or diamonds,clubs or hearts,clubs). Therefore, we have 13*4*12*6=3,744 ways to get a full house and the probability is

	   3,744
	----------- = 0.00144058
	 2,598,960

The reciprocal of this number is 2,598,960/3,744 = 694.

Flush

We have 4 choices for the suit and inside each suite we need to choose 5 cards out of 13. This results in 4*"13-choose-5" choices in total. We also must subtract 36 combinations of 5 cards in sequence (straight flushes). So, the total number of ways to get a flush is:

	     13              13!          13*12*11*10*9
	4 * C   - 36 = 4 * ------- = 4 * --------------- = 5112
	     5              5!*8!         5*4*3*2*1

and the probability is

	   5,112
	----------- = 0.00196694
	 2,598,960

The reciprocal of this number is 2,598,960/5,112 = 508.

Straight

There are 9 possibilites to choose 5 cards in sequence if we ignore suit: 2,3,4,5,6 or 3,4,5,6,7 or ... or 10,J,Q,K,A. Now we have 4 choices for the suit of each card in the sequence and therefore the total of 9*4*4*4*4*4=9216 choices. We also must subtract 36 to account for straight flushes. So, there are 9216-36=9180 ways to get a straight and the probability is

	   9,180
	----------- = 0.00353218
	 2,598,960

The reciprocal of this number is 2,598,960/9,180 = 283.

Three of a Kind

Analogously to the full house hand, there are 13*4=52 ways to choose 3 cards of a kind (of the same rank). The other two cards, have to be one of the 12 other ranks each and they can not form a pair. So, we have "12-choose-2" choices for ranks of these 2 other cards and since the suits of these 2 cards can be arbitrary, we have 4*4 choices for suites. Therefore, there are total of 13*4*66*4*4=54,912 ways to get three of a kind and the probability is

	  54,912
        ----------- = 0.02112845
         2,598,960

The reciprocal of this number is 2,598,960/54,912 = 47.

Two Pairs

There are "13-choose-2" choices for the ranks of pairs, and similarly, we have "4-choose-2" choices for suits in each pair. The last 5th card can be of any rank different from the two ranks of the pairs (11 choices) and of any suit (4 choices). So, there are

	 13    4    4
	C   * C  * C  * 11 * 4 = 78 * 6 * 6 * 11 * 4 = 123,552
	 2     2    2

ways to get two pairs and the probability is

	  123,552
	----------- = 0.04753902
	 2,598,960

The reciprocal of this number is 2,598,960/123,552 = 21.

A Pair

There are 13 choices for the rank of the pair and "4-choose-2"=6 choices for the suits in the pair. Since the other 3 cards must all have different ranks and can be of any suit, we have "12-choose-3"*4*4*4 choices for the cards other than the pair. So, the total number of ways to get a pair is

	      4    12
	13 * C  * C   * 4 * 4 * 4 = 13 * 6 * 220 * 4 * 4 * 4 = 1,098,240
	      2    3

and the probability is

         1,098,240
	----------- = 0.42256903
         2,598,960

The reciprocal of this number is 2,598,960/1,098,240 = 2.37 .