Odds Based on Drawing Outs
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Make on Turn or River Make on River (x × 4)% (x × 3 + 9)% (x × 2)% (x × 2 + (x × 2) ÷ 10)% Outs Actual Est. Error % Error Est. Error % Error Actual Est. Error % Error Est. Error % Error 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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% 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.
This chart is provided courtesy of Wikipedia and is available under the Creative Commons Attribution-ShareAlike License.