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Understanding Standard Deviation in Multiplayer Video Poker
Introduction
This article elaborates on how hands in multi-play video poker are interrelated and how this understanding can be advantageous. By default, a 'hand' refers to the hand drawn rather than the initial deal.
In n-play video poker, the overall variance, where each hand has a variance of v and the covariance between any two hands is c, is calculated as n times v plus n times (n minus 1) times c.
The table below outlines essential data, including the covariance between hands, for six widely played video poker variations.
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 next table displays the variance and standard deviation for both all hands aggregated and individually, specifically for the 9-6 Jacks or Better game.
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
This table provides the variance and standard deviation for both collective hands and individual hands in the 8-5 Bonus Poker variant.
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 reveals the variance and standard deviation for both the total of all hands and for individual hands concerning the 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
This table illustrates the variance and standard deviation for both total hands combined and per individual hand for the 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
In this table, you will find the variance and standard deviation for combined hands as well as for each hand in the 25-15-10-4-4-3 Deuces Wild game.
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 upcoming table reveals variance and standard deviation details for both all hands as a whole as well as per hand for the 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
This table indicates the likelihood of drawing any two specific hands, given the same hand dealt earlier, considering strategies for 9-6 Jacks or Better. The left column denotes 'hand 1' while the top row denotes 'hand 2.' Some probabilities are represented in scientific notation due to their small values.
Probability Pairs Overview 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 subsequent table presents the same data but expands the precision, displaying the combination counts up to 15 digits (the maximum capacity of Excel) for each hand pair, disregarding the order. Notably, the combined outcome for the two hands equals twice the result of a single hand.
Probability Pairs Overview 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 engages in 800 initial hands (in the deal) of 10-play Jacks or Better on a 25¢ machine, wagering five coins each time. What will the standard deviation be for her overall gameplay?
To begin, we will determine the variance per hand in the deal using units. According to the first table, the variance for Jacks or Better is 19.514676 and the covariance is 1.966389.
Thus, the total variance for a hand in the deal is computed as 10 multiplied by 19.514676 plus 10 multiplied by 9 multiplied by 1.966389, resulting in 372.121770 units. This calculation can also be referenced in the Jacks or Better table above.
Next, we will multiply the variance per hand based on the deal by the number of hands Mary played in that deal, which is 800. This results in 800 multiplied by 372.121770, equaling 297697.
Next, we take the overall variance calculated in units and multiply it by the square of the total amount wagered per play, resulting in 297697 multiplied by 1.25.2= $465,152.21.
Lastly, we determine the standard deviation by taking the square root of the variance, which results in $465,152.21.0.5= $682.02.
In the Jacks or Better table, it's noted that the standard deviation per hand in a 10-play scenario is 6.100178.
There are 800×10 = 8000 total hands played.
A general formula for calculating standard deviation is b multiplied by s multiplied by the square root of n, where:
b = bet amount
s = standard deviation per hand
n = number of hands.
Applying this formula provides a combined standard deviation of $1.25 multiplied by 6.100178 multiplied by the square root of 8000, which equals $682.02.