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Last Updated: December 7, 2012

Standard Deviation for Multihand Video Poker

Introduction

This appendix shall answer the question on the standard deviation for n-play video poker for three types of full pay machines. In general the greater the number of plays the greater the standard deviation. In fact the relationship between number of hands and standard deviation is a linear one. The following table shows the breakdown in variance between the deal and the draw.

Video Poker Variance on Deal and Draw

Game Variance on Deal Variance on Draw Total Variance
Full pay deuces wild 3.140053 22.694565 25.834618
10/7 double bonus 3.391375 24.864165 28.255539
9/6 jacks or better 1.966391 17.548285 19.514676
8/5 Bonus poker 2.120027 18.784055 20.904082

The formula for the variance per hand (on the draw) of n-play video poker is n*vardeal+vardraw. The following tables show the variance and standard deviation for all three games and 1, 3, 5, 10, 50, and 100 play. All numbers on a per individual hand basis.

Full Pay Deuces Wild

Full Pay Deuces Wild

n Variance Standard Dev.
1 25.834618 5.082777
3 32.114723 5.666985
5 38.394829 6.196356
10 54.095093 7.354937
50 179.697201 13.405118
100 336.699837 18.349382

10/7 Double Bonus

10/7 Double Bonus

n Variance Standard Dev.
1 28.255539 5.315594
3 35.038288 5.919315
5 41.821037 6.466919
10 58.77791 7.666675
50 194.43289 13.943919
100 364.001616 19.078826

9/6 Jacks or Better

9/6 Jacks or Better

n Variance Standard Dev.
1 19.514676 4.417542
3 23.447459 4.842258
5 27.380241 5.232613
10 37.212196 6.100180
25 66.708000 8.167500
39 94.237534 9.707602
50 115.867841 10.764193
100 214.187396 14.635143

8/5 Bonus Poker

8/5 Bonus Poker

n Variance Standard
Deviation
1 20.904082 4.572098
3 25.144136 5.014393
5 29.38419 5.420719
10 39.984325 6.323316
50 124.785405 11.170739
100 230.786755 15.191667

Let's look at an example. Consider the total standard deviation of 500 initial hands (5000 total hands) of 10-play jacks or better. This would be 50001/2*6.10018 = 431.3479.

Another way to look at this is at the variance and covariance between hands in n-play video poker as follows.

Variance and Covariance Table

Game Variance Covariance
Full pay deuces wild 25.834618 3.140053
10/7 double bonus 28.255539 3.391375
9/6 jacks or better 19.514676 1.966391

To find the total variance of n hands of n-play video poker use the following formula: n*variance+n*(n-1)*covariance. The variance per hand is variance+(n-1)*covariance. The standard deviation per hand is (variance+(n-1)*covariance)1/2.

Sample Problems

For these problems the term "initial hand" refers to the deal (as opposed to the draw). For example if a player played 10-play video poker 5 times he would get 5 initial hands but 50 final hands.

  1. What is the standard deviation of one hand of 1-play jacks or better on a $1 machine with max coins? (answer)
  2. What is the standard deviation of one hand of 1-play jacks or better on a 25 cent machine with max coins? (answer)
  3. What is the standard deviation of 10 hands of 1-play jacks or better on a 25 cent machine with max coins? (answer)
  4. What is the standard deviation of 1 initial hands of 10-play jacks or better on a 25 cent machine with max coins? (answer)
  5. What is the standard deviation of 100 initial hands of 50-play deuces wild on a $5 machine with max coins? (answer)
  6. What is the standard deviation of 50 initial hands of 100-play deuces wild on a $5 machine with max coins? (answer)
  7. What is the standard deviation of 1 initial hand of 8-play 10/7 double bonus on a $2 machine with max coins? (answer)
  8. What is the standard deviation of 2000 initial hands of 23-play double bonus on a $25 machine with max coins? (answer)

Spin Poker

The following table compares the standard deviation per hand on the draw in Spin Poker to multi-play video poker, in 9/6 Jacks or Better. For more information, see my section on Spin Poker.

Standard Deviation

Lines/Plays Spin Poker Multi-Play
2 4.64 4.63
3 4.82 4.84
4 5.4 5.04
5 5.90 5.23
6 6.25 5.42
7 6.55 5.60
8 6.89 5.77
9 7.28 5.94

Recommended Reading

For another good source on this subject visit Jazbo's article An Analysis of N-Play Video Poker. The articles includes variance breakdowns for 13 video poker variations.

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Written by: Michael Shackleford

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