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) 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.
First 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
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
Head-to-head starting hand matchups
When comparing two starting hands, the head-to-head probability describes the likelihood of one hand beating the other after all of the cards have come out. Head-to-head probabilities vary slightly for each particular distinct starting hand matchup, but the approximate average probabilities, as given by Dan Harrington in Harrington on Hold'em [p.125], are summarized in the following table.
Pair vs. 2 undercards 0.83 4.9 : 1
Pair vs. lower pair 0.82 4.5 : 1
Pair vs. 1 overcard, 1 undercard 0.71 2.5 : 1
2 overcards vs. 2 undercards 0.63 1.7 : 1
Pair vs. 2 overcards 0.55 1.2 : 1
These odds are general approximations only derived from averaging all of the hand matchups in each category. The actual head-to-head probabilities for any two starting hands vary depending on a number of factors, including:
- Suited or unsuited starting hands;
- Shared suits between starting hands;
- Connectedness of non-pair starting hands;
- Proximity of card ranks between the starting hands (lowering straight potential);
- Proximity of card ranks toward A or 2 (lowering straight potential);
- Possibility of split pot.
Number of possible hand combinations
1 Opponent 1,225
2 Opponents 690,900
3 Opponents 238,360,500
4 Opponents 56,372,258,250
5 Opponents ˜9.7073 × 1012 (more than 9.7 trillion)
6 Opponents ˜1.2620 × 1015 (more than 1.2 quadrillion)
7 Opponents ˜1.2674 × 1017 (more than 126 quadrillion)
8 Opponents ˜9.9804 × 1018 (almost 10 quintillion)
9 Opponents ˜6.2211 × 1020 (more than 622 quintillion)
The term pot odds in poker refers to the ratio of the current pot size to the amount it costs to call a bet. For example, if you are playing hold'em and the size of the pot is $10 and a player then makes a $2 bet, the pot size includes the bet so the total pot is $12. In order for you to call the bet you have to pay $2 into a $12 pot. You are getting pot odds of 12:2 or 6:1.
Pot odds can be used in many different situations in poker but, at the most basic level, they are useful to determine whether or not a particular draw is profitable over the long run. For example, in hold'em, you have a draw (say four to a flush) that you know is going to come on the next card once out of every five times. This draw is a 4:1 underdog. Say the pot is $20 and you are faced with a $4 bet, you are getting 5:1 pot odds. Because you are getting higher pot odds (5:1) than percentage of the time that your flush will come in (4:1), you can still call here profitably. If the pot was only $5 and the bet was $5 to you, the pot odds here are only 2:1. Given the same draw, it is now unprofitable to call in this situation because of the size of the pot.
In reality, poker situations are usually never quite this simple but, this example shows how a poker player can use pot odds to make the right choice.
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