User:Quaeria/sandbox

Single hand

 * {| class="wikitable" style="font-size: 95%;"

! rowspan=2 | Hand shape ! rowspan=2 | Number of hands ! rowspan=2 | Suit combinations for each hand ! rowspan=2 | Combinations ! colspan=2 | Dealt specific hand ! colspan=2 | Dealt any hand ! Probability !! Odds !! Probability !! Odds
 * Pocket pair || align=center | 13
 * $${4 \choose 2} = 6$$
 * 13 × 6 = 78
 * $$\begin{matrix} \frac{6}{1326} \approx 0.00452 \end{matrix}$$ || 220 : 1
 * $$\begin{matrix} \frac{78}{1326}=\frac{3}{51} \approx 0.0588 \end{matrix}$$ || align=right| 16 : 1
 * Suited cards || align=center | 78
 * $${4 \choose 1} = 4$$
 * 78 × 4 = 312
 * $$\begin{matrix} \frac{4}{1326} \approx 0.00302 \end{matrix}$$ || 331 : 1
 * $$\begin{matrix} \frac{312}{1326}=\frac{12}{51} \approx 0.2353 \end{matrix}$$ || align=right| 3.25 : 1
 * Unsuited cards non paired || align=center | 78
 * $${4 \choose 1}{3 \choose 1} = 12$$
 * 78 × 12 = 936
 * $$\begin{matrix} \frac{12}{1326} \approx 0.00905 \end{matrix}$$ || 110 : 1
 * $$\begin{matrix} \frac{936}{1326}=\frac{36}{51} \approx 0.7059 \end{matrix}$$ || align=right| 0.417 : 1
 * }
 * $$\begin{matrix} \frac{12}{1326} \approx 0.00905 \end{matrix}$$ || 110 : 1
 * $$\begin{matrix} \frac{936}{1326}=\frac{36}{51} \approx 0.7059 \end{matrix}$$ || align=right| 0.417 : 1
 * }


 * {| class="wikitable" style="font-size: 95%;"

!Hand                                                    !! Probability            !! Odds
 * AKs (or any specific suited cards)        ||align=center| 0.00302 ||align=right| 331 : 1
 * AA (or any specific pair)                 ||align=center| 0.00452 ||align=right| 220 : 1
 * AKs, KQs, QJs, or JTs (suited cards)      ||align=center| 0.0121  ||align=right| 81.9 : 1
 * AK (or any specific non-pair incl. suited) ||align=center| 0.0121 ||align=right| 81.9 : 1
 * AA, KK, or QQ                             ||align=center| 0.0136  ||align=right| 72.7 : 1
 * AA, KK, QQ or JJ                          ||align=center| 0.0181  ||align=right| 54.3 : 1
 * Suited cards, jack or better              ||align=center| 0.0181  ||align=right| 54.3 : 1
 * AA, KK, QQ, JJ, or TT                     ||align=center| 0.0226  ||align=right| 43.2 : 1
 * Suited cards, 10 or better                ||align=center| 0.0302  ||align=right| 32.2 : 1
 * Suited connectors                         ||align=center| 0.0392  ||align=right| 24.5 : 1
 * Connected cards, 10 or better             ||align=center| 0.0483  ||align=right| 19.7 : 1
 * Any 2 cards with rank at least queen      ||align=center| 0.0498  ||align=right| 19.1 : 1
 * Any 2 cards with rank at least jack       ||align=center| 0.0905  ||align=right| 10.1 : 1
 * Any 2 cards with rank at least 10         ||align=center| 0.143   ||align=right| 5.98 : 1
 * Connected cards (cards of consecutive rank)||align=center| 0.157  ||align=right| 5.38 : 1
 * Any 2 cards with rank at least 9          ||align=center| 0.208   ||align=right| 3.81 : 1
 * Not connected nor suited, at least one 2-9 ||align=center| 0.534  ||align=right| 0.873 : 1
 * }
 * Suited connectors                         ||align=center| 0.0392  ||align=right| 24.5 : 1
 * Connected cards, 10 or better             ||align=center| 0.0483  ||align=right| 19.7 : 1
 * Any 2 cards with rank at least queen      ||align=center| 0.0498  ||align=right| 19.1 : 1
 * Any 2 cards with rank at least jack       ||align=center| 0.0905  ||align=right| 10.1 : 1
 * Any 2 cards with rank at least 10         ||align=center| 0.143   ||align=right| 5.98 : 1
 * Connected cards (cards of consecutive rank)||align=center| 0.157  ||align=right| 5.38 : 1
 * Any 2 cards with rank at least 9          ||align=center| 0.208   ||align=right| 3.81 : 1
 * Not connected nor suited, at least one 2-9 ||align=center| 0.534  ||align=right| 0.873 : 1
 * }
 * Any 2 cards with rank at least 10         ||align=center| 0.143   ||align=right| 5.98 : 1
 * Connected cards (cards of consecutive rank)||align=center| 0.157  ||align=right| 5.38 : 1
 * Any 2 cards with rank at least 9          ||align=center| 0.208   ||align=right| 3.81 : 1
 * Not connected nor suited, at least one 2-9 ||align=center| 0.534  ||align=right| 0.873 : 1
 * }
 * Not connected nor suited, at least one 2-9 ||align=center| 0.534  ||align=right| 0.873 : 1
 * }
 * }

Head-to-head starting hand matchups

 * {| class="wikitable" style="font-size: 95%;"

! align=left| Favorite-to-underdog matchup !! Probability !! Odds for
 * Pair vs. 2 undercards
 * align=center| 0.83 || align=center| 1 : 4.9
 * Pair vs. lower pair
 * align=center| 0.82 || align=center| 1 : 4.5
 * Pair vs. 1 overcard, 1 undercard
 * align=center| 0.71 || align=center| 1 : 2.5
 * 2 overcards vs. 2 undercards
 * align=center| 0.63 || align=center| 1 : 1.7
 * Pair vs. 2 overcards
 * align=center| 0.55 || align=center| 1 : 1.2
 * }
 * align=center| 0.63 || align=center| 1 : 1.7
 * Pair vs. 2 overcards
 * align=center| 0.55 || align=center| 1 : 1.2
 * }
 * }

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.

For example, vs. is 87.65% to win (0.49% to split), but vs. is 76.81% to win (0.32% to split).

The mathematics for computing all of the possible matchups is simple. However, the computation is tedious to carry out by hand. A computer program can perform a brute force evaluation of the 1,712,304 possible boards for any given pair of starting hands in seconds.

Pocket pairs

 * {| class="wikitable" style="font-size: 95%;"

! Probability of facing one larger pair when holding ! Against 1 !! Against 2 !! Against 3 !! Against 4 !! Against 5 !! Against 6 !! Against 7 !! Against 8 !! Against 9
 * - align=center
 * KK || 0.0049 || 0.0098 || 0.0147 || 1.96% || 0.0244 || 0.0293 || 0.0342 || 0.0391 || 0.0439
 * - align=center
 * QQ || 0.0098 || 0.0195 || 0.0292 || 3.88% || 0.0484 || 0.0579 || 0.0673 || 0.0766 || 0.0859
 * - align=center
 * JJ || 0.0147 || 0.0292 || 0.0436 || 5.77% || 0.0717 || 0.0856 || 0.0992 || 0.1127 || 0.1259
 * - align=center
 * TT || 0.0196 || 0.0389 || 0.0578 || 7.64% || 0.0946 || 0.1124 || 0.1299 || 0.1470 || 0.1637
 * - align=center
 * 99 || 0.0245 || 0.0484 || 0.0718 || 9.46% || 0.1168 || 0.1384 || 0.1593 || 0.1795 || 0.1990
 * - align=center
 * 88 || 0.0294 || 0.0580 || 0.0857 || 11.25% || 0.1384 || 0.1634 || 0.1873 || 0.2101 || 0.2318
 * - align=center
 * 77 || 0.0343 || 0.0674 || 0.0994 || 13.01% || 0.1595 || 0.1874 || 0.2138 || 0.2387 || 0.2619
 * - align=center
 * 66 || 0.0392 || 0.0769 || 0.1130 || 14.73% || 0.1799 || 0.2104 || 0.2389 || 0.2651 || 0.2890
 * - align=center
 * 55 || 0.0441 || 0.0862 || 0.1263 || 16.42% || 0.1996 || 0.2324 || 0.2623 || 0.2892 || 0.3129
 * - align=center
 * 44 || 0.0490 || 0.0956 || 0.1395 || 18.06% || 0.2186 || 0.2532 || 0.2841 || 0.3109 || 0.3334
 * - align=center
 * 33 || 0.0539 || 0.1048 || 0.1526 || 19.67% || 0.2370 || 0.2729 || 0.3040 || 0.3300 || 0.3503
 * - align=center
 * 22 || 0.0588 || 0.1141 || 0.1654 || 21.24% || 0.2546 || 0.2914 || 0.3222 || 0.3464 || 0.3633
 * }

Hands with one ace

 * {| class="wikitable" style="font-size: 95%;"

! Probability of facing an ace with larger kicker when holding ! Against 1 !! Against 2 !! Against 3 !! Against 4 !! Against 5 !! Against 6 !! Against 7 !! Against 8 !! Against 9
 * - align = center
 * AK || 0.00245 || 0.00489 || 0.00733 || 0.00976 || 0.01219 || 0.01460 || 0.01702 || 0.01942 || 0.02183
 * - align = center
 * AQ || 0.01224 || 0.02434 || 0.03629 || 0.04809 || 0.05974 || 0.07126 || 0.08263 || 0.09386 || 0.10496
 * - align = center
 * AJ || 0.02204 || 0.04360 || 0.06468 || 0.08529 || 0.10545 || 0.12517 || 0.14445 || 0.16331 || 0.18175
 * - align = center
 * AT || 0.03184 || 0.06266 || 0.09250 || 0.12139 || 0.14937 || 0.17645 || 0.20267 || 0.22805 || 0.25263
 * - align = center
 * A9 || 0.04163 || 0.08153 || 0.11977 || 0.15642 || 0.19154 || 0.22520 || 0.25745 || 0.28837 || 0.31799
 * - align = center
 * A8 || 0.05143 || 0.10021 || 0.14649 || 0.19038 || 0.23202 || 0.27152 || 0.30898 || 0.34452 || 0.37823
 * - align = center
 * A7 || 0.06122 || 0.11870 || 0.17266 || 0.22331 || 0.27086 || 0.31550 || 0.35741 || 0.39675 || 0.43369
 * - align = center
 * A6 || 0.07102 || 0.13700 || 0.19829 || 0.25523 || 0.30812 || 0.35726 || 0.40291 || 0.44531 || 0.48471
 * - align = center
 * A5 || 0.08082 || 0.15510 || 0.22338 || 0.28615 || 0.34384 || 0.39687 || 0.44561 || 0.49041 || 0.53160
 * - align = center
 * A4 || 0.09061 || 0.17301 || 0.24795 || 0.31609 || 0.37806 || 0.43442 || 0.48567 || 0.53227 || 0.57465
 * - align = center
 * A3 || 0.10041 || 0.19073 || 0.27199 || 0.34509 || 0.41085 || 0.47000 || 0.52322 || 0.57109 || 0.61416
 * - align = center
 * A2 || 0.11020 || 0.20826 || 0.29552 || 0.37315 || 0.44223 || 0.50370 || 0.55840 || 0.60706 || 0.65037
 * }

The flop
The value of a starting hand can change dramatically after the flop. Regardless of initial strength, any hand can flop the nuts&mdash;for example, if the flop comes with three s, any hand holding the fourth has the nuts. Conversely, the flop can undermine the perceived strength of any hand&mdash;a player holding would not be happy to see  on the flop because of the straight and flush possibilities.


 * {| class="wikitable" style="font-size: 95%;"

! rowspan=2 | Board consisting of ! colspan=2 align=center width=150 | Making on flop ! colspan=2 align=center width=150 | Making by turn ! colspan=2 align=center width=150 | Making by river ! Prob. !! Odds  !! Prob. !! Odds  !! Prob. !!  Odds
 * - style="Background: #eee"
 * Three or more of same suit (other suit can have two)
 * align=center| 0.05177 ||align=right| 18.3 : 1
 * align=center| 0.17537 ||align=right| 4.70 : 1
 * align=center| 0.37107 ||align=right| 1.69 : 1
 * Four or more of same suit
 * align=center| 0.01056 ||align=right| 93.7 : 1
 * align=center| 0.04490 ||align=right| 21.3 : 1
 * Rainbow flop (all different suits)
 * align=center| 0.39765 ||align=right| 1.51 : 1
 * align=center| 0.10550 ||align=right| 8.48 : 1
 * Three cards of consecutive rank (but not four consecutive)
 * align=center| 0.03475 ||align=right| 27.8 : 1
 * align=center| 0.10544 ||align=right| 8.48 : 1
 * align=center| 0.19910 ||align=right| 4.02 : 1
 * Four cards to a straight (but not five)
 * align=center| 0.01040 ||align=right| 95.1 : 1
 * align=center| 0.03763 ||align=right| 25.6 : 1
 * Three or more cards of consecutive rank and same suit
 * align=center| 0.00217 ||align=right| 459 : 1
 * align=center| 0.00675 ||align=right| 147 : 1
 * align=center| 0.01305 ||align=right| 75.6 : 1
 * Three of a kind (but not a full house or four of a kind)
 * align=center| 0.00235 ||align=right| 424 : 1
 * align=center| 0.00922 ||align=right| 107 : 1
 * align=center| 0.02113 ||align=right| 46.3 : 1
 * A pair (but not two pair or three or four of a kind)
 * align=center| 0.16941 ||align=right| 4.90 : 1
 * align=center| 0.30425 ||align=right| 2.29 : 1
 * align=center| 0.42257 ||align=right| 1.37 : 1
 * Two pair (but not a full house)
 * align=center| 0.01037 ||align=right| 95.4 : 1
 * align=center| 0.04754 ||align=right| 20.0 : 1
 * }
 * align=center| 0.00922 ||align=right| 107 : 1
 * align=center| 0.02113 ||align=right| 46.3 : 1
 * A pair (but not two pair or three or four of a kind)
 * align=center| 0.16941 ||align=right| 4.90 : 1
 * align=center| 0.30425 ||align=right| 2.29 : 1
 * align=center| 0.42257 ||align=right| 1.37 : 1
 * Two pair (but not a full house)
 * align=center| 0.01037 ||align=right| 95.4 : 1
 * align=center| 0.04754 ||align=right| 20.0 : 1
 * }
 * align=center| 0.01037 ||align=right| 95.4 : 1
 * align=center| 0.04754 ||align=right| 20.0 : 1
 * }

One can see from the table above that more than 60% of the flops will have at least two of the same suit.

Flopping overcards when holding a pocket pair

 * {| class="wikitable" style="font-size: 95%;"

! rowspan=2 | Holding pocket pair ! colspan=2 align=center width=150 | No overcard on flop ! colspan=2 align=center width=150 | No overcard by turn ! colspan=2 align=center width=150 | No overcard by river !      Prob. !!   Odds    !! Prob. !!    Odds     !! Prob. !! Odds
 * - style="Background: #eee"
 * -align=center
 * KK || 0.7745 ||align=right|  0.29 : 1 || 0.7086 ||align=right|      0.41 : 1 || 0.6470 ||align=right|       0.55 : 1
 * -align=center
 * QQ || 0.5857 ||align=right|  0.71 : 1 || 0.4860 ||align=right|      1.06 : 1 || 0.4015 ||align=right|       1.49 : 1
 * -align=center
 * JJ || 0.4304 ||align=right|  1.32 : 1 || 0.3205 ||align=right|      2.12 : 1 || 0.2369 ||align=right|       3.22 : 1
 * -align=center
 * TT || 0.3053 ||align=right|  2.28 : 1 || 0.2014 ||align=right|      3.97 : 1 || 0.1313 ||align=right|       6.61 : 1
 * -align=center
 * 99 || 0.2071 ||align=right|  3.83 : 1 || 0.1190 ||align=right|      7.40 : 1 || 0.0673 ||align=right|      13.87 : 1
 * -align=center
 * 88 || 0.1327 ||align=right|  6.54 : 1 || 0.0649 ||align=right|     14.40 : 1 || 0.0310 ||align=right|      31.21 : 1
 * -align=center
 * 77 || 0.0786 ||align=right| 11.73 : 1 || 0.0318 ||align=right|     30.48 : 1 || 0.0124 ||align=right|      79.46 : 1
 * -align=center
 * 66 || 0.0416 ||align=right| 23.02 : 1 || 0.0133 ||align=right|     74.26 : 1 || 0.0040 ||align=right|     246.29 : 1
 * -align=center
 * 55 || 0.0186 ||align=right| 52.85 : 1 || 0.0043 ||align=right|    229.07 : 1 || 0.0009 ||align=right|   1,057.32 : 1
 * -align=center
 * 44 || 0.0061 ||align=right| 162.33 : 1 || 0.0009 ||align=right| 1,095.67 : 1 || 0.0001 ||align=right|   8,406.78 : 1
 * -align=center
 * 33 || 0.0010 ||align=right| 979.00 : 1 || 0.0001 ||align=right| 15,352.33 : 1 || 0.0000 ||align=right| 353,125.67 : 1
 * }

Notice that there is a better than 35% probability that an ace will come by the river if holding pocket kings, and with pocket queens, the odds are slightly in favor of an ace or a king coming by the turn, and a full 60% in favor of an overcard to the queen by the river. With pocket jacks, there's only a 43% chance that an overcard will not come on the flop and it is better than 3 : 1 that an overcard will come by the river.

Notice, though, that those probabilities would be lower if we consider that at least one opponent happens to hold one of those overcards.

After the flop – outs
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 improving to a winning hand over any unimproved hand held by an opponent. For example, an inside straight draw (e.g. 3-4-6-7 missing the 5 for a straight), and a full house draw (e.g. 6-6-K-K 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


 * $$P = 1 - \left(\frac{47 - outs}{47} \times \frac{46 - outs}{46}\right) = \frac{93outs-outs^2}{2,162}.$$

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


 * {| class="wikitable" style="font-size: 95%;"

! rowspan=2 | Example drawing to ! rowspan=2 | Outs ! colspan=2 width="150" | Make on turn ! colspan=2 width="150" | Make on river ! colspan=2 width="150" | Make on turn or river ! Prob. !! Odds !! Prob. !! Odds !! Prob. !! Odds Inside straight or high pair Inside straight, flush or high pair Open-ended straight, flush or high pair   When drawing to a full house or four of a kind with a pocket pair that has hit a set (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. 
 * - style="Background: #eee"
 * - align=center
 * align=left| Inside straight flush; Four of a kind
 * 1 || 0.0213 || 46.0 : 1 || 0.0217 || 45.0 : 1 || 0.0426 || 22.5 : 1
 * - align=center
 * align=left| Open-ended straight flush; Three of a kind
 * 2 || 0.0426 || 22.5 : 1 || 0.0435 || 22.0 : 1 || 0.0842 || 10.9 : 1
 * - align=center
 * align=left| High pair
 * 3 || 0.0638 || 14.7 : 1 || 0.0652 || 14.3 : 1 || 0.1249 || 7.01 : 1
 * - align=center
 * align=left| Inside straight; Full house
 * 4 || 0.0851 || 10.8 : 1 || 0.0870 || 10.5 : 1 || 0.1647 || 5.07 : 1
 * - align=center
 * align=left| Three of a kind or two pair
 * 5 || 0.1064 || 8.40 : 1 || 0.1087 || 8.20 : 1 || 0.2035 || 3.91 : 1
 * - align=center
 * align=left| Either pair
 * 6 || 0.1277 || 6.83 : 1 || 0.1304 || 6.67 : 1 || 0.2414 || 3.14 : 1
 * - align=center
 * align=left| Full house or four of a kind;
 * 7 || 0.1489 || 5.71 : 1 || 0.1522 || 5.57 : 1 || 0.2784 || 2.59 : 1
 * - align=center
 * align=left| Open-ended straight
 * 8 || 0.1702 || 4.88 : 1 || 0.1739 || 4.75 : 1 || 0.3145 || 2.18 : 1
 * - align=center
 * align=left| Flush
 * 9 || 0.1915 || 4.22 : 1 || 0.1957 || 4.11 : 1 || 0.3497 || 1.86 : 1
 * - align=center
 * align=left| Inside straight or pair
 * 10 || 0.2128 || 3.70 : 1 || 0.2174 || 3.60 : 1 || 0.3839 || 1.60 : 1
 * - align=center
 * align=left| Open-ended straight or high pair
 * 11 || 0.2340 || 3.27 : 1 || 0.2391 || 3.18 : 1 || 0.4172 || 1.40 : 1
 * - align=center
 * align=left| Inside straight or flush; Flush or high pair
 * 12 || 0.2553 || 2.92 : 1 || 0.2609 || 2.83 : 1 || 0.4496 || 1.22 : 1
 * - align=center
 * align=left|
 * 13 || 0.2766 || 2.62 : 1 || 0.2826 || 2.54 : 1 || 0.4810 || 1.08 : 1
 * - align=center
 * align=left| Open-ended straight or pair
 * 14 || 0.2979 || 2.36 : 1 || 0.3043 || 2.29 : 1 || 0.5116 || 0.955 : 1
 * - align=center
 * align=left| Open-ended straight or flush; Flush or pair;
 * 15 || 0.3191 || 2.13 : 1 || 0.3261 || 2.07 : 1 || 0.5412 || 0.848 : 1
 * - align=center
 * align=left|
 * 16 || 0.3404 || 1.94 : 1 || 0.3478 || 1.88 : 1 || 0.5698 || 0.755 : 1
 * - align=center
 * align=left|
 * 17 || 0.3617 || 1.76 : 1 || 0.3696 || 1.71 : 1 || 0.5976 || 0.673 : 1
 * - align=center
 * align=left| Inside straight or flush or pair;
 * 18 || 0.3830 || 1.61 : 1 || 0.3913 || 1.56 : 1 || 0.6244 || 0.601 : 1
 * - align=center
 * align=left|
 * 19 || 0.4043 || 1.47 : 1 || 0.4130 || 1.42 : 1 || 0.6503 || 0.538 : 1
 * - align=center
 * align=left|
 * 20 || 0.4255 || 1.35 : 1 || 0.4348 || 1.30 : 1 || 0.6753 || 0.481 : 1
 * - align=center
 * align=left| Open-ended straight, flush or pair
 * 21 || 0.4468 || 1.24 : 1 || 0.4565 || 1.19 : 1 || 0.6994 || 0.430 : 1
 * }

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 on either the turn or the river. With 20 or more outs, a hand is a better than 2 : 1 favorite to catch at least one out in the two remaining cards.

See the article on pot odds for examples of how these probabilities might be used in gameplay decisions.

Estimating probability of drawing outs - The rule of four and two
Many poker players do not have the mathematical ability to calculate odds in the middle of a poker hand. One solution is to just memorize the odds of drawing outs at the river and turn since these odds are needed frequently for making decisions. Another solution some players use is an easily calculated approximation of the probability for drawing outs, commonly referred to as the "Rule of Four and Two". With two cards to come, the percent chance of hitting x outs is about (x × 4)%. This approximation gives roughly accurate probabilities up to about 12 outs after the flop, with an absolute average error of 0.9%, a maximum absolute error of 3%, a relative average error of 3.5% and a maximum relative error of 6.8%. With one card to come, the percent chance of hitting x is about (x × 2)%. This approximation has a constant relative error of an 8% underestimation, which produces a linearly increasing absolute error of about 1% for each 6 outs.

A slightly more complicated, but significantly more accurate approximation of drawing outs after the flop is to use (x × 4)% for up to 9 outs and (x × 3 + 9)% for 10 or more outs. This approximation has a maximum absolute error of less than 1% for 1 to 19 outs and maximum relative error of less than 5% for 2 to 23 outs. A more accurate approximation for the probability of drawing outs after the turn is (x × 2 + (x × 2) ÷ 10)%. This is easily done by first multiplying x by 2, then rounding the result to the nearest multiple of ten and adding the 10's digit to the first result. For example, 12 outs would be 12 × 2 = 24, 24 rounds to 20, so the approximation is 24 + 2 = 26%. This approximation has a maximum absolute error of less than 0.9% for 1 to 19 outs and a maximum relative error of 3.5% for more than 3 outs. The following shows the approximations and their absolute and relative errors for both methods of approximation.


 * {| class="wikitable" style="font-size: 95%;"

! rowspan=3 | Outs ! colspan=7 width="300" | Make on turn or river ! colspan=7 width="300" | Make on river ! rowspan=2 | Actual ! colspan=3 | (x × 4)% ! colspan=3 | (x × 3 + 9)% ! rowspan=2 | Actual ! colspan=3 | (x × 2)% ! colspan=3 | (x × 2 + (x × 2) ÷ 10)% ! Est. !! Error !! % Error !! Est. !! Error !! % Error !! Est. !! Error !! % Error !! Est. !! Error !! % Error
 * - style="Background: #eee"
 * - style="Background: #eee"
 * - align=right
 * align=center| 1|| 4.2553%|| 4%|| −0.26%|| 6.00%|| 4%||−0.26%||6.00%|| 2.1739%|| 2%||−0.17%||8.00%|| 2%||−0.17%||8.00%
 * - align=right
 * align=center| 2|| 8.4181%|| 8%|| −0.42%|| 4.97%|| 8%||−0.42%||4.97%|| 4.3478%|| 4%||−0.35%||8.00%|| 4%||−0.35%||8.00%
 * - align=right
 * align=center| 3||12.4884%||12%|| −0.49%|| 3.91%||12%||−0.49%||3.91%|| 6.5217%|| 6%||−0.52%||8.00%|| 7%||+0.48%||7.33%
 * - align=right
 * align=center| 4||16.4662%||16%|| −0.47%|| 2.83%||16%||−0.47%||2.83%|| 8.6957%|| 8%||−0.70%||8.00%|| 9%||+0.30%||3.50%
 * - align=right
 * align=center| 5||20.3515%||20%|| −0.35%|| 1.73%||20%||−0.35%||1.73%||10.8696%||10%||−0.87%||8.00%||11%||+0.13%||1.20%
 * - align=right
 * align=center| 6||24.1443%||24%|| −0.14%|| 0.60%||24%||−0.14%||0.60%||13.0435%||12%||−1.04%||8.00%||13%||−0.04%||0.33%
 * - align=right
 * align=center| 7||27.8446%||28%|| +0.16%|| 0.56%||28%||+0.16%||0.56%||15.2174%||14%||−1.22%||8.00%||15%||−0.22%||1.43%
 * - align=right
 * align=center| 8||31.4524%||32%|| +0.55%|| 1.74%||32%||+0.55%||1.74%||17.3913%||16%||−1.39%||8.00%||18%||+0.61%||3.50%
 * - align=right
 * align=center| 9||34.9676%||36%|| +1.03%|| 2.95%||36%||+1.03%||2.95%||19.5652%||18%||−1.57%||8.00%||20%||+0.43%||2.22%
 * - align=right
 * align=center|10||38.3904%||40%|| +1.61%|| 4.19%||39%||+0.61%||1.59%||21.7391%||20%||−1.74%||8.00%||22%||+0.26%||1.20%
 * - align=right
 * align=center|11||41.7206%||44%|| +2.28%|| 5.46%||42%||+0.28%||0.67%||23.9130%||22%||−1.91%||8.00%||24%||+0.09%||0.36%
 * - align=right
 * align=center|12||44.9584%||48%|| +3.04%|| 6.77%||45%||+0.04%||0.09%||26.0870%||24%||−2.09%||8.00%||26%||−0.09%||0.33%
 * - align=right
 * align=center|13||48.1036%||52%|| +3.90%|| 8.10%||48%||−0.10%||0.22%||28.2609%||26%||−2.26%||8.00%||29%||+0.74%||2.62%
 * - align=right
 * align=center|14||51.1563%||56%|| +4.84%|| 9.47%||51%||−0.16%||0.31%||30.4348%||28%||−2.43%||8.00%||31%||+0.57%||1.86%
 * - align=right
 * align=center|15||54.1166%||60%|| +5.88%||10.87%||54%||−0.12%||0.22%||32.6087%||30%||−2.61%||8.00%||33%||+0.39%||1.20%
 * - align=right
 * align=center|16||56.9843%||64%|| +7.02%||12.31%||57%||+0.02%||0.03%||34.7826%||32%||−2.78%||8.00%||35%||+0.22%||0.62%
 * - align=right
 * align=center|17||59.7595%||68%|| +8.24%||13.79%||60%||+0.24%||0.40%||36.9565%||34%||−2.96%||8.00%||37%||+0.04%||0.12%
 * - align=right
 * align=center|18||62.4422%||72%|| +9.56%||15.31%||63%||+0.56%||0.89%||39.1304%||36%||−3.13%||8.00%||40%||+0.87%||2.22%
 * - align=right
 * align=center|19||65.0324%||76%||+10.97%||16.86%||66%||+0.97%||1.49%||41.3043%||38%||−3.30%||8.00%||42%||+0.70%||1.68%
 * - align=right
 * align=center|20||67.5301%||80%||+12.47%||18.47%||69%||+1.47%||2.18%||43.4783%||40%||−3.48%||8.00%||44%||+0.52%||1.20%
 * - align=right
 * align=center|21||69.9352%||84%||+14.06%||20.11%||72%||+2.06%||2.95%||45.6522%||42%||−3.65%||8.00%||46%||+0.35%||0.76%
 * - align=right
 * align=center|22||72.2479%||88%||+15.75%||21.80%||75%||+2.75%||3.81%||47.8261%||44%||−3.83%||8.00%||48%||+0.17%||0.36%
 * - align=right
 * align=center|23||74.4681%||92%||+17.53%||23.54%||78%||+3.53%||4.74%||50.0000%||46%||−4.00%||8.00%||51%||+1.00%||2.00%
 * }

Either of these approximations is generally accurate enough to aid in most pot odds calculations.

Runner-runner outs
Some outs for a hand require drawing an out on both the turn and the river—making two consecutive outs is called a runner-runner. Examples would be needing two cards to make a straight, flush, or three or four of a kind. Runner-runner outs can either draw from a common set of outs or from disjoint sets of outs. Two disjoint outs can either be conditional or independent events.

Common outs
Drawing to a flush is an example of drawing from a common set of outs. Both the turn and river need to be the same suit, so both outs are coming from a common set of outs—the set of remaining cards of the desired suit. After the flop, if $$x$$ is the number of common outs, the probability $$P$$ of drawing runner-runner outs in Texas hold 'em is


 * $$P = \frac{x}{47} \times \frac{x-1}{46}= \frac{x^2-x}{2,162}.$$

Since a flush would have 10 outs, the probability of a runner-runner flush draw is $$\begin{matrix} \frac{10}{47} \times \frac{9}{46} = \frac{90}{2162} \approx \frac{1}{24} \approx 0.04163 \end{matrix}$$. Other examples of runner-runner draws from a common set of outs are drawing to three or four of a kind. When counting outs, it is convenient to convert runner-runner outs to "normal" outs (see "After the flop"). A runner-runner flush draw is about the equivalent of one "normal" out.

The following table shows the probability and odds of making a runner-runner from a common set of outs and the equivalent normal outs.


 * {| class="wikitable" style="font-size: 95%;"

! Likely drawing to  !! Common outs !! Probability !! Odds !! Equivalent outs Inside-only straight flush
 * Four of a kind (with pair)
 * Four of a kind (with pair)
 * align=center| 2 || align=center| 0.00093 || align=right| 1,080 : 1 || align=center| 0.02
 * Three of a kind (with no pair)
 * align=center| 3 || align=center| 0.00278 || align=right|   359 : 1 || align=center| 0.07
 * align=center| 4 || align=center| 0.00556 || align=right|   179 : 1 || align=center| 0.13
 * align=center| 5 || align=center| 0.00925 || align=right|   107 : 1 || align=center| 0.22
 * Two pair or three of a kind (with no pair)
 * align=center| 6 || align=center| 0.01388 || align=right|  71.1 : 1 || align=center| 0.33
 * align=center| 7 || align=center| 0.01943 || align=right|  50.5 : 1 || align=center| 0.46
 * align=center| 8 || align=center| 0.02590 || align=right|  37.6 : 1 || align=center| 0.61
 * align=center| 9 || align=center| 0.03330 || align=right|  29.0 : 1 || align=center| 0.78
 * Flush
 * align=center| 10 || align=center| 0.04163 || align=right| 23.0 : 1 || align=center| 0.98
 * }
 * align=center| 7 || align=center| 0.01943 || align=right|  50.5 : 1 || align=center| 0.46
 * align=center| 8 || align=center| 0.02590 || align=right|  37.6 : 1 || align=center| 0.61
 * align=center| 9 || align=center| 0.03330 || align=right|  29.0 : 1 || align=center| 0.78
 * Flush
 * align=center| 10 || align=center| 0.04163 || align=right| 23.0 : 1 || align=center| 0.98
 * }
 * align=center| 9 || align=center| 0.03330 || align=right|  29.0 : 1 || align=center| 0.78
 * Flush
 * align=center| 10 || align=center| 0.04163 || align=right| 23.0 : 1 || align=center| 0.98
 * }
 * Flush
 * align=center| 10 || align=center| 0.04163 || align=right| 23.0 : 1 || align=center| 0.98
 * }

Disjoint outs
Two outs are disjoint when there are no common cards between the set of cards needed for the first out and the set of cards needed for the second out. The outs are independent of each other if it does not matter which card comes first, and one card appearing does not affect the probability of the other card appearing except by changing the number of remaining cards; an example is drawing two cards to an inside straight. The outs are conditional on each other if the number of outs available for the second card depends on the first card; an example is drawing two cards to an outside straight.

After the flop, if $$x$$ is the number of independent outs for one card and $$y$$ is the number of outs for the second card, then the probability $$P$$ of making the runner-runner is


 * $$P = \frac{x}{47} \times \frac{y}{46} \times 2 = \frac{xy}{1081}.$$

For example, a player holding after the flop  needs a  and either a  or  on the turn and river to make a straight. There are 4 s and 8 s and s, so the probability is $$\begin{matrix}\frac{4 \times 8}{1081} \approx 0.0296\end{matrix}$$.

The probability of making a conditional runner-runner depends on the condition. For example, a player holding after the flop  can make a straight with,  or. The number of outs for the second card is conditional on the first card—a or  (8 cards) on the first card leaves only 4 outs ( or, respectively) for the second card, while a  or  (8 cards) for the first card leaves 8 outs ( { , } or { , }, respectively) for the second card. The probability $$P$$ of a runner-runner straight for this hand is calculated by the equation


 * $$P = \left(\frac{8}{47} \times \frac{4}{46}\right) + \left(\frac{8}{47} \times \frac{8}{46}\right) = \frac{96}{2162} \approx 0.0444$$

The following table shows the probability and odds of making a runner-runner from a disjoint set of outs for common situations and the equivalent normal outs.


 * {| class="wikitable" style="font-size: 95%;"

! Drawing to                              !! Probability !! Odds  !! Equivalent outs
 * - align=center
 * align=left| Outside straight             || 0.04440 ||align=right| 21.5 : 1 || 1.04
 * - align=center
 * align=left| Inside+outside straight      || 0.02960 ||align=right| 32.8 : 1 || 0.70
 * - align=center
 * align=left| Inside-only straight         || 0.01480 ||align=right| 66.6 : 1 || 0.35
 * - align=center
 * align=left| Outside straight flush       || 0.00278 ||align=right|  359 : 1 || 0.07
 * - align=center
 * align=left| Inside+outside straight flush || 0.00185 ||align=right| 540 : 1 || 0.04
 * }

The preceding table assumes the following definitions.


 * Outside straight and straight flush
 * Drawing to a sequence of three cards of consecutive rank from 3-4-5 to 10-J-Q where two cards can be added to either end of the sequence to make a straight or straight flush.


 * Inside+outside straight and straight flush
 * Drawing to a straight or straight flush where one required rank can be combined with one of two other ranks to make the hand. This includes sequences like 5-7-8 which requires a 6 plus either a 4 or 9 as well as the sequences J-Q-K, which requires a 10 plus either a 9 or A, and 2-3-4 which requires a 5 plus either an A or 6.


 * Inside-only straight and straight flush
 * Drawing to a straight or straight flush where there are only two ranks that make the hand. This includes hands such as 5-7-9 which requires a 6 and an 8 as well as A-2-3 which requires a 4 and a 5.

Compound outs
The strongest runner-runner probabilities lie with hands that are drawing to multiple hands with different runner-runner combinations. These include hands that can make a straight, flush or straight flush, as well as four of a kind or a full house. Calculating these probabilities requires adding the compound probabilities for the various outs, taking care to account for any shared hands. For example, if $$P_s$$ is the probability of a runner-runner straight, $$P_f$$ is the probability of a runner-runner flush, and $$P_{sf}$$ is the probability of a runner-runner straight flush, then the compound probability $$P$$ of getting one of these hands is


 * $$P = P_s + P_f - P_{sf}.$$

The probability of the straight flush is subtracted from the total because it is already included in both the probability of a straight and the probability of a flush, so it has been added twice and must therefore be subtracted from the compound outs of a straight or flush.

The following table gives the compound probability and odds of making a runner-runner for common situations and the equivalent normal outs.


 * {| class="wikitable" style="font-size: 95%;"

! Drawing to                                                 !! Probability !! Odds  !! Equivalent outs
 * - align=center
 * align=left| Flush, outside straight or straight flush       || 0.08326 ||align=right| 11.0 : 1 || 1.98
 * - align=center
 * align=left| Flush, inside+outside straight or straight flush || 0.06938 ||align=right| 13.4 : 1 || 1.65
 * - align=center
 * align=left| Flush, inside-only straight or straight flush   || 0.05550 ||align=right| 17.0 : 1 || 1.30
 * }

Some hands have even more runner-runner chances to improve. For example, holding the hand after a flop of  there are several runner-runner hands to make at least a straight. The hand can get two cards from the common outs of {J, Q} (5 cards) to make a full house or four of a kind, can get a J (2 cards) plus either a 7 or 10 (6 cards) to make a full house from these independent disjoint outs, and is drawing to the compound outs of a flush, outside straight or straight flush. The hand can also make {7, 7} or {10, 10} (each drawing from 3 common outs) to make a full house, although this will make four of a kind for anyone holding the remaining 7 or 10 or a bigger full house for anyone holding an overpair. Working from the probabilities from the previous tables and equations, the probability $$P$$ of making one of these runner-runner hands is a compound probability


 * $$P = 0.08326 + 0.00925 + \frac{2 \times 6}{1081} + (0.00278 \times 2) \approx 0.1092$$

and odds of 8.16 : 1 for the equivalent of 2.59 normal outs. Almost all of these runner-runners give a winning hand against an opponent who had flopped a straight holding 8, 9, but only some give a winning hand against (this hand makes bigger flushes when a flush is hit) or against  (this hand makes bigger straights when a straight is hit with ). When counting outs, it is necessary to adjust for which outs are likely to give a winning hand—this is where the skill in poker becomes more important than being able to calculate the probabilities.