# Variance in Fixed Limit Holdem

• Bronze
Joined: 27.09.2008
Hi!

I promised prepare article about variance in Fixed Limit Holdem. Because it seems I cannot do it in one shot I will start thread and will update the post later.

Normal Distribution

Many things that are affected by random factors behave according to Normal distribution. It does not matter if you are manfacturer of some part with required lenght or if you are waiting for bus that should arrive every 5 minuts.

Both of these examples will behave according to picture above. Only difference will be,
• that part is manufactured for lenght 1m (center of the distribution) +- 1mm standard deviation (Sigma) and
• bus will arrive every 5 minutes (center of the distribution) +- 20 seconds standard devition (Sigma).

The picture shows how many data will fit in interval from the center line.
For
• +- 1Sigma it is about 70% of all occurences - bus from previous example will arrive by 70% in interval 4 minutes 40s to 5 minuts and 20 seconds, this is general and normal behaviour.
• +- 2Sigma it is about 95.5% of tall occurences - so 19 times of 20 bus will arrive in interval from 4 minutes 20 seconds to 5 minutes and 40 seconds, we might assume that bus was delayed due to long stay on red light or driver was speeding.
• +- 3sigma it is about 99.97% all occurences - by this we can say bus will arrive every time with in interval of 4 minutes to 6 minutes. If does not arrive in this interval something special had to occure - e. g. flat tire.

We can apply same principles on poker game. In Fixed Limit Holdem we know that good player will have average income 1-2 BB (Big Bets) per 100 hands. At the same time we know that variance in Fixed Limit FR Holdem is about 15 BB per 100 hands.

Therefore players results per 100 hands will be
• by 70% in interval -14BB to +16BB
• by 95.5% in interval -29BB to +31BB,
• by by 99.97% in interval -44BB to +46BB
• in 0.03% he will lose more that 44BB or gain more than 46BB (naturally 1 time in 3333 cases)

Confidence Intervals
Two following pictures shows and the confidence interval in which should player stay. Note that x axis shows number of hands, y axis shows number of BB.

The narrow line is average income in BB per number of hands.

The inner lines shows 95.5% (2 standard deviation) confidence interval, the outer lines shows 99.97% (3 standard deviation) interval - player should never move outside these outer borders.

Tabels list some exact numbers for 100, 1 000, 10 000 and 100 000 hands.

[Center] FL FR 1BB/100 hand, 15BB / 100 hand stdev
Good FL Player[/center]

[Center] FL FR 2BB/100 hand, 15BB / 100 hand stdev
Excelent FL Player[/center]

Both picuters shows, that player are heavily affected by variance and even after 90 000 hands good player might be still breakeven (2stdev).

However excelent player is affected by variance less because he will be mostlikely in profit in just 25 000 hands (97.5%).

Example:
From Jan 2nd to Jan 5th I played 4145 hands with winrate 4.24BB/ 100 hands.

Assume that stdev is 15BB / 100 hands. Therefore for 4145 hand the standard deviation of win rate is 2.33 BB/ 100 hands.

How big is chance that I am not losing player on my limit?

My winrate is 1.81 stdev (4.24/2.33) above being break even.
Outside (one sided) of 1.81 stdev lies 3.5% of values.

That gives me 96.5% confidence that I am winning player.

Second question would be how big is probability that my longterm winrate is bigger than 1BB/100 hands?

(4.24-1)/2.33 = 1.39 stdev -> distance in stdev between my current winrate and winrate I would like to achive.

Outside (one sided) of 1.39 stdev lies 8.2% of values.
That gives me 91.8% chance that my long term winrate is bigger than 1BB/100 hand.

The last question is, do I beat my limit with winrate at least 2BB/100 hands?
(4.24-2)/2.33 = 0.96 stdev

Outside (one sided) of 1.39 stdev lies 16.8% of values.

There might be chance about 83% that I will beat my level with winrate bigger than 2BB/100 hands.

What will follow?
• explanation how accurate is your win rate based on number of hands played
• anything that I will be able answer in there will be any questions

TTT
• 12 replies
• Black
Joined: 05.02.2008
I have no idea how did you get those numbers tbh, but this is what I found in one DC video:

There's uncertainty in BB/100 which is counted as "Standard deviation"/SQRT("number of hands/100")
Confidence is counted as follows:
68% represents winrate +/- 1*uncertainty
95% represents winrate +/- 2*uncertainty
99% represents winrate +/- 3*uncertainty

Your data (4145 hands, 4.24 winrate, 15 SD) give this output:
with 68% confidence is your true winrate between 1.91 and 6.57 BB/100
with 95% confidence is your true winrate between -0.42 and 8.90 BB/100
with 99% confidence is your true winrate between -2.75 and 11.23 BB/100
(we're interested only in bottom value obv.)

According to this result, you're not winning player with 95% confidence, because of possible variance etc. (You'd be at least b/e after 5k hands).

Your calculation can be more accurate than the one i mentioned after, let's say, 30k+ hands (or maybe after 100k+) I think... Or not?

btw: Your graph says I still can be excellent player at 3/6 even I'm -60BB after 45k hands; thank you for encouragement
• Bronze
Joined: 25.11.2008
So i have played 9023 hands and my BB/100 is 3.95/100.

According to the charts that u have presented what does it all mean?
I play FL full ring .5/1.
• Bronze
Joined: 02.07.2008
wow solid post, tyvm!

can you make that graphs with 3d surfaces? i think that may help to understand it, (or at least make it look very COOL ).
• Bronze
Joined: 27.09.2008
Opal,

The limits that you showed are discussed in the beggining of the post
1, 2 and 3 standard deviations (Sigma in Normal Distribution Chart), corresponding amount of date that lies between the numbers are approx. 70%, 95.5%, 99.97% as you mentioned.

+- 3 stdev is considered natural process regulation boundaries, because probability that you by outside (one sided is about 0.0015 or 0.15%)

The numbers that you present are right. I computed only probability that my true winrate is about 0BB / 100 hands, 1BB / 100 hands and 2BB / hands - in that case you substract centrain number of values that lies on one side of distribution after specific number of standard deviation.

See how the computation is done in detail:

Assume that SW tells you, that you stdev is 15BB in 100 hands and your winrate is allready mention 4.24 BB/ 100 hands.

So you have normal distribution N100(mu, sigma^2) = N100(4.24, 15*15).
Second parametr of distribution is square of standard deviation - variance.
That is distribution that comes for 100 hands. To get distribution for 4145 hands you have to sum 41.25 of these distribution.

Therefore
41.25 * N100(4.24, 15*15) = N4125(4.24*4125, 41.25*15*15) = N4125 (174.9, 9281).

Have a look here how can you get these formula
Sum of normally distributed random variables

So my mean winning in 4125 hands is 174.9BB with 96.3BB/4125 hands standard deviation.

Now I can normalize these numbers to 100 by just dividing both of them by 41.25 and I will get that based on my sample size my long term win rate is 4.24BB/ 100 hands with standard deviation of 2.33 BigBlinds / 100 hands.

And now I can have a look to Z table and see how much data lies outside ceintairn number of standard deviations from the center of the distribution (one sided). A that are the numbers I have presented.

To be break even my winrate has to be >0BB/100 hands. My winrate is 4.24 with standard deviation of 2.33. Thet gives me 1.81 of standard deviation of center of the distribution. In interval (1.81, inf) there is only 3.5% of all values in these distribution. Therefore I have confidence 96.5% percent that I am not in this interval, so I am long term winning player.

That number 2.33 BB/ 100 hands is not my variance in 100 hands, it is variance how accurately is my winrate for 100 hands set. If I play my variance in 100 hands is still 15BB / 100 hands.

In the evening I will post table that will show how acurate is your winrate after certain number of hands. E.g. from 1 000 to 1 000 000 hands.

TTT

P.S.: After 45k of hands (Short Handed) the standard deviation of winrate is 0,85 BB / 100 hands... so probability that your winrate is >1BB/100 hands is about 14%.
• Bronze
Joined: 27.09.2008
Originally posted by saywhat
So i have played 9023 hands and my BB/100 is 3.95/100.

According to the charts that u have presented what does it all mean?
I play FL full ring .5/1.
Your winrate is 3.95BB/100 hands with standard deviation of 1.58BB

Actual winrate > 0 BB - 99.4%
Actual winrate > 1 BB - 96.8%
Actual winrate > 2 BB - 89%
Actual winrate > 3 BB - 72%
Actual winrate > 4 BB - 48%

TTT
• Gold
Joined: 04.06.2008
Hi TTT i think your post is very helpfull but I have to mention that these results need to be taken with a pinch of salt to alot of beginners who may be reading this post (already over 100 views so far).

Other factors come into account esp. for beginners in that people play differently (hopefully better) after awhile, this is sometimes known as the learning curve.

Also internet poker is increasingly becoming a harder place to make profit mainly due to sites such as this, and that 100k hands can take many months to do in which your oppenents usually change.

Im sure that you already take this into account, im just posting this for other users who might read it and take your post as gospel truth.
• Bronze
Joined: 27.09.2008
Hi!

I would agree. For real beginner such analysis probably does not make any sense at all. The learning curve is very steep and data in couple of thousands hands would not be valid at all (significant changes in VPIP, PFR, AF, postflop readining skills, etc.)

This would rather make sense for player that played several thousands hands and would like to decide whether stay at the limit he is currently playing or move back to the previous one.

As somewhere mentioned player will be moving up in limits until he reach the limit that he cannot beat and will stay there... and that is not the point. Point is to make money.

So this may help analyse risk/reward ratio with limited sample size (probably with about only tens of thousands hands on limit).

Let's say I have decision that I can play either \$1/\$2 FL with winrate 2BB/100 hands or I can play FL \$2/\$4 with winrate 1BB / 100 hands. In general and in long term the expected results will be same (not counting rakeback). But I will have to deal with much bigger downswings on higher limit.

So unless I have some reason (play against tough opponents to improve my game) I would stick to \$1/\$2 level where I have much better consistency in my winnings.

TTT
• Bronze
Joined: 25.03.2005
Hi TTT241,

very nice post. Looks very interesting.

In real I think varaiance can be even worse. I saw winning players lose 1000BB for example. (Is that one one the 0,03% cases?)

I somehow don't understand how it can be possible to get a 84% probabilty that you win more than 2BB/100 as it's highly unlikely to win more than that in real.
Is that because of the sample sample size? I think I'm missing something here. :/

If you win the same at a higher limit I would try to go for the higher to get used to the better opponents if you plan to move up in the future.
• Black
Joined: 10.09.2007
I remember a (german) site about this topic, perhaps you can find it in the german forums cirith?
• Bronze
Joined: 25.03.2005
Originally posted by Yoghi
I remember a (german) site about this topic, perhaps you can find it in the german forums cirith?
You know how big the german page is.

Maybe later.
• Bronze
Joined: 27.09.2008
The winrate is not true anymore... just got 100BB downswing . In about 1500 hands - I did not win any big pot, when I hit hand all other players folded to turn bet or I was outdrawn on River (e.g. small fullhouse to KK on turn and river or nut flush to 5 full of 6 - already in forum), also in that sample size my winrate at showdown got like 10% down with about same frequency of seing the it as before.

ciRith, there are still some assumtion - like 15BB. May be, my game is variable and I am playing with like 18BB, than the range will be much wider, I was really lucky and hit my to 3% in this sample size, etc.

And still, you need to consider that anything in +-2 stdev is possible result.

Let's see how will be my results by the end of the month - I will be hapy for 1BB/hands ). And hopefully will have bankroll for 1/2 (at least 500BB).

TTT
• Bronze
Joined: 25.03.2005
Originally posted by TTT241
The winrate is not true anymore... just got 100BB downswing . In about 1500 hands - I did not win any big pot, when I hit hand all other players folded to turn bet or I was outdrawn on River (e.g. small fullhouse to KK on turn and river or nut flush to 5 full of 6 - already in forum), also in that sample size my winrate at showdown got like 10% down with about same frequency of seing the it as before.

ciRith, there are still some assumtion - like 15BB. May be, my game is variable and I am playing with like 18BB, than the range will be much wider, I was really lucky and hit my to 3% in this sample size, etc.

And still, you need to consider that anything in +-2 stdev is possible result.

Let's see how will be my results by the end of the month - I will be hapy for 1BB/hands ). And hopefully will have bankroll for 1/2 (at least 500BB).

TTT
Good luck with that.