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Texas Hold’em – After the flop

During play—that is, from the flop and onwards—drawing probabilities come down to a question of outs. All situations which have the same number of outs have the same probability of winning. For example, an inside straight draw (e.g. 34 67 missing the 5 for a straight), and a full house draw (e.g. 66KK drawing for one of the pairs to become three-of-a-kind) are equivalent. Each can be satisfied by four cards—four 5s in the first case, and the other two 6s and other two kings in the second.

The probabilities of drawing these outs are easily calculated. At the flop there remain 47 unseen cards, so the probability is (outs ÷ 47). At the turn there are 46 unseen cards so the probability is (outs ÷ 46). The cumulative probability of making a hand on either the turn or river can be determined as the complement of the odds of not making the hand on the turn and not on the river. The probability of not drawing an out is (47 − outs) ÷ 47 on the turn and (46 − outs) ÷ 46 on the river; taking the complement of these conditional probabilities gives the probability of drawing the out by the river which is calculated by the formula

P = 1 – (((47 – outs)/47) x ((46-outs)/46))

For reference, the probability and odds for some of the more common numbers of outs are given here.

Likely drawing to Outs Make on turn Make on river Make on turn or river
Prob. Odds Prob. Odds Prob. Odds
Inside straight flush; Four of a kind 1 0.0213 46.0 : 1 0.0217 45.0 : 1 0.0426 22.5 : 1
Open-ended straight flush; Three of a kind 2 0.0426 22.5 : 1 0.0435 22.0 : 1 0.0842 10.9 : 1
High pair 3 0.0638 14.7 : 1 0.0652 14.3 : 1 0.1249 7.01 : 1
Inside straight; Full house 4 0.0851 10.8 : 1 0.0870 10.5 : 1 0.1647 5.07 : 1
Three of a kind or two pair 5 0.1064 8.40 : 1 0.1087 8.20 : 1 0.2035 3.91 : 1
Either pair 6 0.1277 6.83 : 1 0.1304 6.67 : 1 0.2414 3.14 : 1
Full house or four of a kind; (see note)
Inside straight or high pair
7 0.1489 5.71 : 1 0.1522 5.57 : 1 0.2784 2.59 : 1
Open-ended straight 8 0.1702 4.88 : 1 0.1739 4.75 : 1 0.3145 2.18 : 1
Flush 9 0.1915 4.22 : 1 0.1957 4.11 : 1 0.3497 1.86 : 1
Inside straight or pair 10 0.2128 3.70 : 1 0.2174 3.60 : 1 0.3839 1.60 : 1
Open-ended straight or high pair 11 0.2340 3.27 : 1 0.2391 3.18 : 1 0.4172 1.40 : 1
Inside straight or flush; Flush or high pair 12 0.2553 2.92 : 1 0.2609 2.83 : 1 0.4496 1.22 : 1
13 0.2766 2.62 : 1 0.2826 2.54 : 1 0.4810 1.08 : 1
Open-ended straight or pair 14 0.2979 2.36 : 1 0.3043 2.29 : 1 0.5116 0.955 : 1
Open-ended straight or flush; Flush or pair;
Inside straight, flush or top pair
15 0.3191 2.13 : 1 0.3261 2.07 : 1 0.5412 0.848 : 1
16 0.3404 1.94 : 1 0.3478 1.88 : 1 0.5698 0.755 : 1
17 0.3617 1.76 : 1 0.3696 1.71 : 1 0.5976 0.673 : 1
Inside straight or flush or pair;
Open-ended straight, flush or high pair
18 0.3830 1.61 : 1 0.3913 1.56 : 1 0.6244 0.601 : 1
19 0.4043 1.47 : 1 0.4130 1.42 : 1 0.6503 0.538 : 1
20 0.4255 1.35 : 1 0.4348 1.30 : 1 0.6753 0.481 : 1
Open-ended straight, flush or pair 21 0.4468 1.24 : 1 0.4565 1.19 : 1 0.6994 0.430 : 1
  • Note: When drawing to a full house or four of a kind with a pocket pair that has hit trips (three of a kind) on the flop, there are 6 outs to get a full house by pairing the board and one out to make four of a kind. This means that if the turn does not pair the board or make four of a kind, there will be 3 additional outs on the river, for a total of 10, to pair the turn card and make a full house. This makes the probability of drawing to a full house or four of a kind on the turn or river 0.334 and the odds are 1.99 : 1. This makes drawing to a full house or four of a kind by the river about 8½ outs.

It is worth noting in the preceding table that if a player doesn’t fold before the river, a hand with at least 14 outs after the flop has a better than 50% chance to catch one of its outs by the river. With 20 or more outs, a hand is a better than 2 : 1 favorite to catch at least one out by the river.

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

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