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Standard Deviation for Multihand Video Poker
Introduction
This article indicates the covariance between hands in multi-play video poker and how to use such information. Except where stated, a "hand" shall refer to a hand on the draw, as opposed to the deal.
The Total variance in n-play video poker, where the variance per hand is v, and the covariance between any two hands is c, equals n×v + n×(n-1)×c.
The following table shows basic information, including covariance between any two hands, in six common video poker games.
Basic Statistics
| Game | Pay Table | Return | Variance | Covariance |
|---|---|---|---|---|
| Bonus Deuces | 9-4-4-3 | 0.994502 | 32.662818 | 3.806094 |
| Bonus Poker | 8-5 | 0.991660 | 20.904082 | 2.120027 |
| Deuces Wild | 25-15-10-4-4-3 | 0.994179 | 25.679180 | 3.024385 |
| Double Bonus | 9-7-5 | 0.991065 | 28.547130 | 3.350788 |
| Double Double Bonus | 9-6 | 0.989808 | 41.984981 | 4.809024 |
| Jacks or Better | 9-6 | 0.995439 | 19.514676 | 1.966389 |
Jacks or Better
The following table shows the variance and standard deviation, for both all hands combined as well as per hand, for 9-6 Jacks or Better
9-6 Jacks or Better
| Plays | Total Variance |
Total Standard Deviation |
Per Play Variance |
Per Play Standard Deviation |
|---|---|---|---|---|
| 1 | 19.514676 | 4.417542 | 19.514676 | 4.417542 |
| 3 | 70.342362 | 8.387035 | 23.447454 | 4.842257 |
| 5 | 136.901160 | 11.700477 | 27.380232 | 5.232612 |
| 10 | 372.121770 | 19.290458 | 37.212177 | 6.100178 |
| 25 | 1667.700300 | 40.837486 | 66.708012 | 8.167497 |
| 50 | 5793.386850 | 76.114301 | 115.867737 | 10.764188 |
| 100 | 21418.718700 | 146.351354 | 214.187187 | 14.635135 |
Bonus Poker
The following table shows the variance and standard deviation, for both all hands combined as well as per hand, for 8-5 Bonus Poker
8-5 Bonus Poker
| Plays | Total Variance |
Total Standard Deviation |
Per Play Variance |
Per Play Standard Deviation |
|---|---|---|---|---|
| 1 | 20.904082 | 4.572098 | 20.904082 | 4.572098 |
| 3 | 75.432408 | 8.685183 | 25.144136 | 5.014393 |
| 5 | 146.920950 | 12.121095 | 29.384190 | 5.420719 |
| 10 | 399.843250 | 19.996081 | 39.984325 | 6.323316 |
| 25 | 1794.618250 | 42.362935 | 71.784730 | 8.472587 |
| 50 | 6239.270250 | 78.989051 | 124.785405 | 11.170739 |
| 100 | 23078.675500 | 151.916673 | 230.786755 | 15.191667 |
Double Bonus Poker
The following table shows the variance and standard deviation, for both all hands combined as well as per hand, for 9-7-5 Double Bonus Poker
9-7-5 Double Bonus Poker
| Plays | Total Variance |
Total Standard Deviation |
Per Play Variance |
Per Play Standard Deviation |
|---|---|---|---|---|
| 1 | 28.547130 | 5.342951 | 28.547130 | 5.342951 |
| 3 | 105.746118 | 10.283293 | 35.248706 | 5.937062 |
| 5 | 209.751410 | 14.482797 | 41.950282 | 6.476904 |
| 10 | 587.042220 | 24.228954 | 58.704222 | 7.661868 |
| 25 | 2724.151050 | 52.193400 | 108.966042 | 10.438680 |
| 50 | 9636.787100 | 98.167139 | 192.735742 | 13.882930 |
| 100 | 36027.514200 | 189.809152 | 360.275142 | 18.980915 |
Double Double Bonus Poker
The following table shows the variance and standard deviation, for both all hands combined as well as per hand, for 9-6 Double Double Bonus Poker
9-6 Double Double Bonus Poker
| Plays | Total Variance |
Total Standard Deviation |
Per Play Variance |
Per Play Standard Deviation |
|---|---|---|---|---|
| 1 | 41.984981 | 6.479582 | 41.984981 | 6.479582 |
| 3 | 154.809087 | 12.442230 | 51.603029 | 7.183525 |
| 5 | 306.105385 | 17.495868 | 61.221077 | 7.824390 |
| 10 | 852.661970 | 29.200376 | 85.266197 | 9.233970 |
| 25 | 3935.038925 | 62.729889 | 157.401557 | 12.545978 |
| 50 | 13881.357850 | 117.819174 | 277.627157 | 16.662147 |
| 100 | 51807.835700 | 227.613347 | 518.078357 | 22.761335 |
Deuces Wild
The following table shows the variance and standard deviation, for both all hands combined as well as per hand, for 25-15-10-4-4-3 Deuces Wild
25-15-10-4-4-3 Deuces Wild
| Plays | Total Variance |
Total Standard Deviation |
Per Play Variance |
Per Play Standard Deviation |
|---|---|---|---|---|
| 1 | 25.679180 | 5.067463 | 25.679180 | 5.067463 |
| 3 | 95.183850 | 9.756221 | 31.727950 | 5.632757 |
| 5 | 188.883600 | 13.743493 | 37.776720 | 6.146277 |
| 10 | 528.986450 | 22.999705 | 52.898645 | 7.273145 |
| 25 | 2456.610500 | 49.564206 | 98.264420 | 9.912841 |
| 50 | 8693.702250 | 93.240025 | 173.874045 | 13.186131 |
| 100 | 32509.329500 | 180.303437 | 325.093295 | 18.030344 |
Bonus Deuces Wild
The following table shows the variance and standard deviation, for both all hands combined as well as per hand, for 9-4-4-3 Bonus Deuces Wild
9-4-4-3 Bonus Deuces Wild
| Plays | Total Variance |
Total Standard Deviation |
Per Play Variance |
Per Play Standard Deviation |
|---|---|---|---|---|
| 1 | 32.662818 | 5.715139 | 32.662818 | 5.715139 |
| 3 | 120.825018 | 10.992043 | 40.275006 | 6.346259 |
| 5 | 239.435970 | 15.473719 | 47.887194 | 6.920057 |
| 10 | 669.176640 | 25.868449 | 66.917664 | 8.180322 |
| 25 | 3100.226850 | 55.679681 | 124.009074 | 11.135936 |
| 50 | 10958.071200 | 104.680806 | 219.161424 | 14.804102 |
| 100 | 40946.612400 | 202.352693 | 409.466124 | 20.235269 |
Probability Pairs
The following table shows the probability of any two specific hands on the draw, given the same hand on the deal, assuming strategy for 9-6 Jacks or Better. The left row shows "hand 1" and the top column shows "hand 2." Due to the very small probabilities in some fields, I use scientific notation.
Probability Pairs Table 1 — 9-6 Jacks or Better
| Hand 1 | Nothing | JoB | 2 Pair | 3 Kind | Straight | Flush | F.H. | 4 Kind | S.F. | R.F. |
|---|---|---|---|---|---|---|---|---|---|---|
| Nothing | 3.77E-01 | 7.93E-02 | 4.53E-02 | 2.85E-02 | 5.42E-03 | 6.66E-03 | 2.40E-03 | 6.11E-04 | 6.99E-05 | 1.42E-05 |
| Jacks or better | 7.93E-02 | 9.92E-02 | 2.01E-02 | 1.28E-02 | 9.38E-04 | 9.26E-04 | 1.09E-03 | 2.78E-04 | 6.93E-06 | 5.68E-06 |
| Two pair | 4.53E-02 | 2.01E-02 | 5.12E-02 | 7.92E-03 | 1.08E-04 | 8.13E-05 | 4.40E-03 | 1.87E-04 | 1.21E-06 | 4.98E-07 |
| Three of a kind | 2.85E-02 | 1.28E-02 | 7.92E-03 | 2.25E-02 | 4.33E-05 | 3.13E-05 | 1.65E-03 | 9.38E-04 | 4.25E-07 | 1.85E-07 |
| Straight | 5.42E-03 | 9.38E-04 | 1.08E-04 | 4.33E-05 | 4.65E-03 | 6.03E-05 | 2.45E-06 | 3.34E-07 | 5.52E-06 | 7.96E-07 |
| Flush | 6.66E-03 | 9.26E-04 | 8.13E-05 | 3.13E-05 | 6.03E-05 | 3.25E-03 | 1.38E-06 | 1.93E-07 | 9.94E-06 | 1.66E-06 |
| Full house | 2.40E-03 | 1.09E-03 | 4.40E-03 | 1.65E-03 | 2.45E-06 | 1.38E-06 | 1.91E-03 | 6.66E-05 | 7.41E-09 | 6.85E-09 |
| Four of a kind | 6.11E-04 | 2.78E-04 | 1.87E-04 | 9.38E-04 | 3.34E-07 | 1.93E-07 | 6.66E-05 | 2.82E-04 | 9.86E-10 | 8.61E-10 |
| Straight flush | 6.99E-05 | 6.93E-06 | 1.21E-06 | 4.25E-07 | 5.52E-06 | 9.94E-06 | 7.41E-09 | 9.86E-10 | 1.54E-05 | 3.69E-08 |
| Royal flush | 1.42E-05 | 5.68E-06 | 4.98E-07 | 1.85E-07 | 7.96E-07 | 1.66E-06 | 6.85E-09 | 8.61E-10 | 3.69E-08 | 1.71E-06 |
The next table presents the same information, but to more significant digits. It shows the number of combinations to 15 digits (the maximum of Excel) for each pair of hands, without regard to order. Note the total return of the two combined hands equals two times the return for one hand.
Probability Pairs Table 2 — 9-6 Jacks or Better
| Hand 1 | Hand 2 | Combinations | Probability | Pays | Return |
|---|---|---|---|---|---|
| Nothing | Nothing | 57,664,992,337,108,000,000 | 0.377187 | 0 | 0.000000 |
| Nothing | Jacks or better | 24,232,729,458,658,400,000 | 0.158506 | 1 | 0.158506 |
| Nothing | Two pair | 13,845,304,964,002,300,000 | 0.090562 | 2 | 0.181124 |
| Nothing | Three of a kind | 8,726,039,157,387,020,000 | 0.057077 | 3 | 0.171231 |
| Nothing | Straight | 1,657,016,578,993,360,000 | 0.010839 | 4 | 0.043354 |
| Nothing | Flush | 2,035,490,553,224,720,000 | 0.013314 | 6 | 0.079885 |
| Nothing | Full house | 734,942,598,554,528,000 | 0.004807 | 9 | 0.043265 |
| Nothing | Four of a kind | 186,860,795,577,763,000 | 0.001222 | 25 | 0.030556 |
| Nothing | Straight flush | 21,359,122,264,576,200 | 0.000140 | 50 | 0.006986 |
| Nothing | Royal flush | 4,338,415,760,266,080 | 0.000028 | 800 | 0.022702 |
| Jacks or better | Jacks or better | 15,165,995,951,987,900,000 | 0.099201 | 2 | 0.198402 |
| Jacks or better | Two pair | 6,140,587,770,092,040,000 | 0.040166 | 3 | 0.120497 |
| Jacks or better | Three of a kind | 3,915,849,147,073,900,000 | 0.025614 | 4 | 0.102454 |
| Jacks or better | Straight | 286,715,798,957,348,000 | 0.001875 | 5 | 0.009377 |
| Jacks or better | Flush | 283,137,319,731,984,000 | 0.001852 | 7 | 0.012964 |
| Jacks or better | Full house | 332,470,711,745,820,000 | 0.002175 | 10 | 0.021747 |
| Jacks or better | Four of a kind | 84,953,934,410,987,400 | 0.000556 | 26 | 0.014448 |
| Jacks or better | Straight flush | 2,119,322,635,042,600 | 0.000014 | 51 | 0.000707 |
| Jacks or better | Royal flush | 1,735,582,704,176,590 | 0.000011 | 801 | 0.009093 |
| Two pair | Two pair | 7,831,401,262,721,210,000 | 0.051225 | 4 | 0.204901 |
| Two pair | Three of a kind | 2,420,196,605,329,560,000 | 0.015831 | 5 | 0.079153 |
| Two pair | Straight | 33,016,723,781,798,200 | 0.000216 | 6 | 0.001296 |
| Two pair | Flush | 24,847,188,037,349,400 | 0.000163 | 8 | 0.001300 |
| Two pair | Full house | 1,344,465,032,419,130,000 | 0.008794 | 11 | 0.096736 |
| Two pair | Four of a kind | 57,039,536,401,736,600 | 0.000373 | 27 | 0.010074 |
| Two pair | Straight flush | 369,632,440,017,432 | 0.000002 | 52 | 0.000126 |
| Two pair | Royal flush | 152,242,916,946,336 | 0.000001 | 802 | 0.000799 |
| Three of a kind | Three of a kind | 3,444,111,124,875,160,000 | 0.022528 | 6 | 0.135168 |
| Three of a kind | Straight | 13,253,848,139,056,700 | 0.000087 | 7 | 0.000607 |
| Three of a kind | Flush | 9,579,178,876,536,860 | 0.000063 | 9 | 0.000564 |
| Three of a kind | Full house | 503,473,320,786,464,000 | 0.003293 | 12 | 0.039519 |
| Three of a kind | Four of a kind | 286,901,966,781,062,000 | 0.001877 | 28 | 0.052546 |
| Three of a kind | Straight flush | 129,844,380,330,888 | 0.000001 | 53 | 0.000045 |
| Three of a kind | Royal flush | 56,538,398,938,368 | 0.000000 | 803 | 0.000297 |
| Straight | Straight | 711,149,591,709,176,000 | 0.004652 | 8 | 0.037213 |
| Straight | Flush | 18,447,113,220,812,200 | 0.000121 | 10 | 0.001207 |
| Straight | Full house | 750,203,629,122,672 | 0.000005 | 13 | 0.000064 |
| Straight | Four of a kind | 102,194,252,051,088 | 0.000001 | 29 | 0.000019 |
| Straight | Straight flush | 1,686,711,113,699,520 | 0.000011 | 54 | 0.000596 |
| Straight | Royal flush | 243,362,705,981,664 | 0.000002 | 804 | 0.001280 |
| Flush | Flush | 496,154,126,958,398,000 | 0.003245 | 12 | 0.038944 |
| Flush | Full house | 421,220,447,825,760 | 0.000003 | 15 | 0.000041 |
| Flush | Four of a kind | 58,944,675,640,320 | 0.000000 | 31 | 0.000012 |
| Flush | Straight flush | 3,039,629,528,763,520 | 0.000020 | 56 | 0.001113 |
| Flush | Royal flush | 507,089,614,448,808 | 0.000003 | 806 | 0.002673 |
| Full house | Full house | 291,555,196,668,645,000 | 0.001907 | 18 | 0.034327 |
| Full house | Four of a kind | 20,376,082,044,866,200 | 0.000133 | 34 | 0.004532 |
| Full house | Straight flush | 2,265,084,537,408 | 0.000000 | 59 | 0.000001 |
| Full house | Royal flush | 2,094,928,008,912 | 0.000000 | 809 | 0.000011 |
| Four of a kind | Four of a kind | 43,043,223,890,517,600 | 0.000282 | 50 | 0.014077 |
| Four of a kind | Straight flush | 301,525,772,352 | 0.000000 | 75 | 0.000000 |
| Four of a kind | Royal flush | 263,216,361,648 | 0.000000 | 825 | 0.000001 |
| Straight flush | Straight flush | 2,352,314,821,359,550 | 0.000015 | 100 | 0.001539 |
| Straight flush | Royal flush | 11,282,026,370,328 | 0.000000 | 850 | 0.000063 |
| Royal flush | Royal flush | 261,652,407,890,112 | 0.000002 | 1600 | 0.002738 |
| Total | 0 | 152,881,798,431,626,000,000 | 1.000000 | 1.990878 |
Example Problems
Mary plays 800 initial hands (on the deal) of 10-play Jacks or Better. The game is a 25¢ machine and Mary bet five coins per hand. What is the standard deviation of all her play?
First, let's find the variance per hand on the deal in units. As the first table shows, the variance in Jacks or Better is 19.514676 and the covariance is 1.966389.
The total variance per hand on the deal is 10*19.514676 + 10*9*1.966389 = 372.121770 units. This figure can also be found in the Jacks or Better table above.
Second, multiply the variance per hand on deal by the number of hands played (on the deal), which is 800. That gives us 800*372.121770 = 297697.
Third, multiply the total variance in units by the amount bet per play squared. That gives us 297697 * 1.252 = $465,152.21.
Finally, take the square root of the variance to get the standard deviation: $465,152.210.5 = $682.02.
Note in the Jacks or Better table the standard deviation per hand in 10-play is 6.100178.
There are 800×10 = 8000 total hands played.
A general formula for standard deviation is b × s × sqrt(n), where:
b = bet amount
s = standard deviation per hand
n = number of hands.
Using that formula, we get a total standard deviation of $1.25 × 6.100178 × sqrt(8000) = $682.02.