Big Starting Hand -vs- Big Starting Hand - How often should you see them?

This article tries to give the average poker player a visual picture of how often big hands (AA -vs- KK, AA -vs- QQ and KK -vs-QQ) should come up during a poker session. To analyze this I used the DealGuardian card shuffling and dealing engine to simulate a number of hands. For each hand the following starting hand counts were recorded: AA, KK, QQ, JJ, TT, AA-vs-KK, AA-vs-QQ and KK-vs-QQ. Afterwords, the statistics of how often you should see these particular hand combinations are calculated, not just as percentages, but as a result of frequency over time.

This article will leave out most of the math used. There are a number of other sites dedicated to showing starting hand percentages and odds, so I will leave it up to the reader to search those sites.

To start with I setup the data gathering software to simulate 9 handed poker games using the DealGuardian engine. Only the pre-flop player hole cards were dealt and the information stored for the analysis. Each trial simulated 8 hours of play based on a 70 hand/hour rate. The results of the 100,000 trial analysis was used to write this article:

Trials:..................100,000
Simulated Hours:.........800,000
Hands Per Hour:...............70
Total Hands:..........56,000,000

Let me first start by validating the randomness of the DealGuardian engine by showing the results of the big pairs (AA, KK, QQ, JJ and TT) and compare theme to the expected results.

The probability of a player being dealt any pair for the starting two cards in a hold'em game is 0.45% (4 and ½ tenths of a percent http://en.wikipedia.org/wiki/Poker_probability_%28Texas_hold_%27em%29). In a nine handed game, this number is increased to 9 X 0.00453 to give the probability any player can have a specific pair. The result is that any player being dealt AA in a 9 handed game is approximately 4.08%. After running the 100,000 trials, the data demonstrated the following percentages for big pairs using DealGuardian:

AA:.............4.07%
KK:.............4.07%
QQ:.............4.07%
JJ:.............4.07%
TT:.............4.08%

This should be evidence that the test engine is sufficient for simulating an unbiased shuffle and distribution of cards. Now on to the bigger question about data gathered about big pair -vs- big pair.


We will next look at the percentages of this situation gathered from the same 100,000 trials. I will then translate he percentages into hourly rates to demonstrate how often a player should witness these situations when playing poker. After 56,000,000 hands the following probability of a having big pair -vs- big pair is:

AA -vs- KK:.............0.16%
AA -vs- QQ:.............0.16%
KK -vs- QQ:.............0.16%

To get an approximation of how close this is to the expected value you can multiply the 1 player pair percentage to an 8 player pair percentage to get 0.15%. That's a very small percentage of time overall. The following data is how many times big pair situations came up in the 56,000,000 hand:

AA -vs- KK Total:.............88,771
AA -vs- QQ Total:.............88,637
KK -vs- QQ Total:.............88,692

That is after 800,000 hours of simulated play. So how often should a player witness this scenario at a 9 handed table? Well, the rate I got for each situation is around 0.11/hour which means you should see any one of these once every 9 hours and either of them once every 3 hours. That's a pretty low rate compared to what I think I'm experiencing online. Which is why I started this analysis in the first place.

My next effort will be to collect online statistics to see if these numbers line up. It's going to be difficult gathering 56 million hands of legitimate online poker room data, but I feel its will only take a small number of hands to see if the online stats are in line with these results.

Please feel free to share these results as often as you like, as long as they are not altered. A link to the PDF version can be found at:

http://www.securecarddealer.com/presentations/BigHandAnalysisReport.pdf