# Something about variance and expected results

• Bronze
Joined: 27.09.2008
Hi!

I liked very nice thread about variance in SNGs. So I decided explore variance more in NL cashgames.

In some of the article on PokerPortal.cz I found that solid TAG player will have expected income about 10BB per 100 hands with about 60 to 120BB standard deviation.

Because the results in poker will behave according to normal distribution, we have normal distribution N(10, 10 000) for stdev = 100 (N(mean, variance = stdev^2))

If we have a look on the following picture it shows how are results distributed according to mean (mu), and standard deviation Sigma:

Therefore in 100 hands:
You will finish -90BB to 110BB with probability 70% (+- 1stdev)
You will finish -190BB to 210BB with probability 95.4% (+- 2stdev)
You will finish -290BB to 310BB with probability 99.97% (+- 3stdev)

What will happen if sum hands up and we played about 1 000 hands?
Based on this we will get new normal distribution with the following parameters - N(10*10, 10*10000) = N(100, 100 000), that gives stdev =
316BB

I put to the table expected results in number of Buy Ins (BI) for standard deviation of size 100

The following image show expected results (in BB) in 120 000 hands with confidence intervals of 95% [-2stdev, +2stdev] and 99.97% [-3stdev, +3stdev]. The [-3stdev, +3stdev] are commonly considered as natural regulation boundaries of process and you should not play outside these boundaries. Be outside 3 stdev is mostlikely caused by something else than natural variability (e.g. tilt).

In 10 000 hands:
• Probability that you will lose more than 5BI in 1000 hands is about 40%
• Probability that you will be in red numbers is about 16% (1 in 6 cases)

Generaly
• there is 3% probability that you will be in lost greater than 10 BI
• you should not be in lost greater than 22BI (minimum of 3stdev - green curve))
• after 40k hands there is 97% chance that you will not suffer lost

Update
The following list shows how accurate is your winrate, again 95% confidence intervals (+- 2stdev) are set:
• 1 000 hands - WinRate +- 62BB
• 10 000 hands - Winrate +- 20 BB
• 100 000 hands - Winrate +- 6.4BB
• 300 000 hands - Winrate +- 3.6BB
• 1 000 000 hands - Winrate +- 2 BB

I hope that this will help you understand how big is variability in poker. Even after 100 000 played hands the size of natural variability is approximately same as expected income.

Regards

TTT
• 8 replies
• Bronze
Joined: 26.05.2007
Very nice post.
• Bronze
Joined: 27.09.2008
Post update:

Added accuracy of Winrate according to number of hands.

TTT
• Bronze
Joined: 20.04.2008
• Bronze
Joined: 27.09.2008
It is one in general SNG discussion - in Top 3 articles

TTT
• Bronze
Joined: 06.07.2008
I am statistician and I must say the standart deviation in floating(and is unforcastable), so you can't apply this model, GG. Even for a programmed bot it's floating. Also mean is floating. It's very, very complicated.

Also, you can't maintain it's normal destribution and you cannot check, it's impossible to build model for poker winnings.
• Bronze
Joined: 20.04.2008
Thatnks TTT241 for posting the link! And thank you Balanarm for bringing this thread back up otherwise I probably wouldn't have noticed..
• Bronze
Joined: 27.09.2008
Sure, its floating... but if you cannot do some shortcuts you will not move anywhere (like fixed stdev, assumtion about winrate, etc.) ...

And I think normal distribution is quite close to reality... like in all other processes where some random factors are included...

TTT
• Bronze
Joined: 07.05.2008
This is just scary.