Poker Probability (Texas Hold 'em) - Starting Hands

Starting Hands

The probability of being dealt various starting hands can be explicitly calculated. In Texas Hold 'em, a player is dealt two down (or hole or pocket) 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 in which you receive the first two cards is not significant, the 2,652 permutations are divided by the 2 ways of ordering two cards.) Alternatively, the number of possible starting hands is represented as the binomial coefficient

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		 Suit combinations
for each hand		 Combinations Dealt specific hand Dealt any hand
Probability Odds Probability Odds
Pocket pair 13 13 × 6 = 78 220 : 1 16 : 1
Suited cards 78 78 × 4 = 312 331 : 1 3.25 : 1
Unsuited cards non paired 78 78 × 12 = 936 110 : 1 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) 0.00452 220 : 1
AKs, KQs, QJs, or JTs (suited cards) 0.0121 81.9 : 1
AK (or any specific non-pair incl. suited) 0.0121 81.9 : 1
AA, KK, or QQ 0.0136 72.7 : 1
AA, KK, QQ or JJ 0.0181 54.3 : 1
Suited cards, jack or better 0.0181 54.3 : 1
AA, KK, QQ, JJ, or TT 0.0226 43.2 : 1
Suited cards, 10 or better 0.0302 32.2 : 1
Suited connectors 0.0392 24.5 : 1
Connected cards, 10 or better 0.0483 19.7 : 1
Any 2 cards with rank at least queen 0.0498 19.1 : 1
Any 2 cards with rank at least jack 0.0905 10.1 : 1
Any 2 cards with rank at least 10 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

Read more about this topic:  Poker Probability (Texas Hold 'em)

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