Probability for starting hands in Texas Hold 'em

The probability of being dealt various starting hands can be explicitly calculated. In Texas Hold ’em, a player is dealt two down (or hole) cards. The first card can be any one of 52 playing cards in the deck and the second card can be any one of the 51 remaining cards. This gives 52 × 51 ÷ 2 = 1,326 possible starting hand combinations. (Since the order of the cards is not significant, the 2,652 combinations are divided by the 2 ways of ordering two cards.) Alternately, the number of possible starting hands is represented as the binomial coefficient

(52/2) = 1,326

which is the number of possible combinations of choosing 2 cards from a deck of 52 playing cards.

The 1,326 starting hands can be reduced for purposes of determining the probability of starting hands for Hold ’em—since suits have no relative value in poker, many of these hands are identical in value before the flop. The only factors determining the strength of a starting hand are the ranks of the cards and whether the cards share the same suit. Of the 1,326 combinations, there are 169 distinct starting hands grouped into three shapes: 13 pocket pairs (paired hole cards), 13 × 12 ÷ 2 = 78 suited hands and 78 unsuited hands; 13 + 78 + 78 = 169. The relative probability of being dealt a hand of each given shape is different. The following shows the probabilities and odds of being dealt each type of starting hand.

Hand shape	Number of hands	Permutations for each hand	Combinations
Pocket pair	13	(4/2) = 6	13 × 6 = 78
Suited cards	78	(4/1) = 4	78 × 4 = 312
Unsuited cards	78	(4/1)(3/1) = 12	78 × 12 = 936

Hand shape	Dealt specific hand		Dealt any hand
Hand shape	Probability	Odds	Probability	Odds
Pocket pair	6/1326 ~ 0.00453	220 : 1	78/1326 ~ 0.0588	16 : 1
Suited cards	4/1326 ~ 0.00302	331 : 1	312/1326 ~ 0.2353	3.25 : 1
Unsuited cards	12/1326 ~ 0.00905	110 : 1	936/1326 ~ 0.7059	0.417 : 1

Here are the probabilities and odds of being dealt various other types of starting hands.

Hand	Probability	Odds
AKs (or any specific suited cards)	0.00302	331 : 1
AA (or any specific pair)	0.00453	220 : 1
AKs, KQs, QJs, or JTs	0.0121	81.9 : 1
AK (or any specific non-pair)	0.0121	81.9 : 1
AA, KK, or QQ	0.0136	72.7 : 1
Suited cards, J or better	0.0181	54.3 : 1
AA, KK, QQ, JJ, or TT	0.0226	43.2 : 1
Suited cards, T or better	0.0302	32.2 : 1
Suited connectors	0.0392	24.5 : 1
Connected cards, T or better	0.0483	19.7 : 1
Any 2 cards with rank at least Q	0.0498	19.1 : 1
Any 2 cards with rank at least J	0.0905	10.1 : 1
Any 2 cards with rank at least T	0.143	5.98 : 1
Connected cards (cards of consecutive rank)	0.157	5.38 : 1
Any 2 cards with rank at least 9	0.208	3.81 : 1
Not connected nor suited, at least one 2-9	0.534	0.873 : 1