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Last Updated: January 21, 2019

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?

The answer is $682.02

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.

 

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