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View Full Version : Plotting a distribution of expected session outcomes.

bisonbison
01-19-2005, 11:57 PM
I've got an idea but not enough knowledge of excel to know if it's possible, and I was thinking you prob guys are probably the ones to ask.

Basically, using the SD equation from Homer's streaks thread, I'd like to be able to plot a bell curve of outcomes from X sessions of Y hands.

In other words, given that my win-rate is BlahBlahBlah and my SD is poop, what are the range of session outcomes I can expect to see over 10k sessions?

sthief09
01-20-2005, 12:14 AM
[ QUOTE ]
I've got an idea but not enough knowledge of excel to know if it's possible, and I was thinking you prob guys are probably the ones to ask.

Basically, using the SD equation from Homer's streaks thread, I'd like to have to be able to plot a bell curve of outcomes from X sessions of Y hands.

In other words, given that my win-rate is BlahBlahBlah and my SD is poop, what are the range of session outcomes I can expect to see over 10k sessions?

[/ QUOTE ]

I tried to search for it, but to no avail. Homer posted an Excel spreadsheet that does exactly this recently. try PM'ing him, or maybe your search skills are better than mine. he posted it in the zoo

bisonbison
01-20-2005, 12:31 AM
I found it:

I'm not sure if this does exactly what I'm after but it sure is neat.

gaming_mouse
01-20-2005, 01:39 AM

Hope it wasn't prompted by a downswing.

gm

bisonbison
01-20-2005, 01:41 AM
meh, I don't believe in downswings.

That guy
01-21-2005, 04:02 PM
I am no stats guru but if memory serves me right:

If you have the Std Dev and you can assume 'normality' in the distribution,

If you are +2 BB's per session (consisting of say 95 hands)
you can run a simple confidence interval where;

+1 SD = 2bb's + 18bb's
-1 SD = 2bb's - 18bb's

playing at 3/6 =
12 + 108 = +\$120
12 - 108 = -\$96

Any session should fall within a -\$96 to +\$120 band, with ~68% confidence

You have 84% confidence that your next session will not lose more than \$-96. Thus you will lose more than -\$96 about 16% of the time.

95% of sessions should fall within 1.65 standard deviations
(-\$166 to +\$190).

yoshi_yoshi
01-21-2005, 08:22 PM
I'm not sure about excel, but you can easily plot the graph on a graphing calculator or a computer program if you have one.

The formula is:

P(x)= [e^(-((x - EV)^2)/(2*SD^2))] / (SD * sqrt(2*pi))

You should get a nice looking bell shaped curve, centered at your EV.

Little Lew
01-22-2005, 01:55 PM
Confessing some experience with experimental stats in prior life ---------

I don't think that you can make any assumption of normality in the distribution of the bb/hr or bb/session figures because, unless the player is incredibly disciplined, the player will react to losing or winning streaks by changing behavior- perhaps to limit losses or pressing to maximize gains.

Since you have all the the session figures, it would be optimal to actually chart all the session figures and see if these figures are actually distributed normally. Once you are convinced that the figures are normally distributed then you can have confidence in the estimates of possible per session gains/losses.

A link to a discussion on skewness (the degree of asymetry of a distribution) or kurtosis ( the excess peakedness or flatness of a distribution) is at

overtechnical blather on kurtosis/skewness (http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm)

My intuition is that my own results would be greatly skewed because I am perfectly happy to win as much as I can (on the rare occasions I actually win) but I tend to tighten up, sulk and even leave when my losses get too high at any session. (Actually it is fear of my spuse looking over my shoulder and then clutching my neck.) /images/graemlins/smile.gif

That guy
01-22-2005, 05:34 PM
Monte Carlo analysis do not need the assumption of normality so the original post is very valid.

Analyst
01-24-2005, 01:41 PM
[ QUOTE ]
I've got an idea but not enough knowledge of excel to know if it's possible, and I was thinking you prob guys are probably the ones to ask.

Basically, using the SD equation from Homer's streaks thread, I'd like to be able to plot a bell curve of outcomes from X sessions of Y hands.

In other words, given that my win-rate is BlahBlahBlah and my SD is poop, what are the range of session outcomes I can expect to see over 10k sessions?

[/ QUOTE ]

Interesting - I was just thinking about this topic this morning after a massively bad run last night. The magnitude of upswings and downswings (mainly the latter) is a frequent topic on the HUSH board.

The key issue, I think, is that session results are ultra-short runs and may very well not fall into a normal distribution. Yes, when you add them up in the long run, the law of large numbers takes over and your total results will be approximately normally distributed. However, as we all know the long run is, well, long - perhaps hundreds of thousands of hands or even more while a single session is very short indeed (well, it seemed pretty long last night. /images/graemlins/frown.gif )

Basically, I'm beginning to think it is completely unrealistic to say "My win rate is X/100 hands with a SD of Y/100 hands, so I should expect ~2/3 of my 100-hand sessions to fall in the range of X +/- Y."

To address your question, yes, I think there is a way to model this with Excel using Monte Carlo simulation. Complex and tedious, but worthwhile study and I'm thinking about how to structure it. A bigger hand database than mine would be helpful - want to collaberate?

Pokerscott
01-24-2005, 03:13 PM
[ QUOTE ]
The key issue, I think, is that session results are ultra-short runs and may very well not fall into a normal distribution. Yes, when you add them up in the long run, the law of large numbers takes over and your total results will be approximately normally distributed. However, as we all know the long run is, well, long - perhaps hundreds of thousands of hands or even more while a single session is very short indeed (well, it seemed pretty long last night. )

[/ QUOTE ]

I did a simulation analysis for this. The results for cumulative bankroll and ROI are pretty darn normal even after 50-100 SnGs.

results tables (http://forumserver.twoplustwo.com/showflat.php?Cat=&amp;Number=1570111&amp;page=6&amp;view=colla psed&amp;sb=5&amp;o=14&amp;fpart=1)

Pokerscott

That guy
01-24-2005, 09:04 PM
Assuming the math in Homers spreadsheet is correct, that is an excellent tool to simulate poker. Poker is a long series of % plays.

The key question though is what is a useful assumption for standard deviation. Homer used 18 BB's which seems reasonable. More often than not, I fall within a band of +18 to -18 BB's per table in an hour of play... and it makes sense to me that a +36 or -36 BB session (1 hour) would be a '2 standard deviation event' (ie, occuring &lt;5% of the time).

Play with his spreadsheet with higher deviations to 'stress test' it... After doing this for a while, my read on it is that there is a high statistical likelihood that at some point you are going to run into something like a -120 BB downswing... and perhaps -200 or maybe even -300 BB's... A really bad run isn't a certainty by any means but it wouldn't be uncommon to do this. This is just the nature of playing a long series of hands which are each subject to the laws of statistics similar to flipping a coin many times will undoubtedly have some crazy runs of heads... throw in laying down the best hand a couple of times during a bad run and this run can get ugly fast...

Also whenever your standard deviation (call it 20 BB's/hour) is high relative to the average earn (call it 2 BB's/hour), you call this 'mean blur' and it basically says that you should take your hourly rate with a grain of salt when thinking about forecasting sessions... There is a lot of short-term luck (variance) in the game, even by excellent players.

(also note in Homers simulations that quite often there is NOT a terrible run and that you may simply be the recipient of good luck -- in addition to good play)...

Derek in NYC
01-25-2005, 11:07 AM
Excel has a histogram function but you need the stats pack add on. if you look under Tools and you dont see a Data Analysis option, you need the addon. Go here if you need it: Microsoft (http://office.microsoft.com/en-us/assistance/HP051985541033.aspx)

Analyst
01-25-2005, 05:30 PM
Interesting stuff! However, I'm not sure that it addresses quite what Bison was asking (or at least what I think he was asking), nor quite what my model would analyze - the very, very short run of a single session (say, 100-400 hands). Specifically, what is the distribution of these short run outcomes? I suspect, and with no analysis or modelling yet it's only a suposition, that these outcomes may not be distributed normally.

That guy
01-25-2005, 08:30 PM
of course 100-400 hands won't be normally distributed. statistics describes a long data series... it is non-sensical to talk about statistics over 100 hands..

uuDevil
01-25-2005, 10:10 PM
[ QUOTE ]
Interesting stuff! However, I'm not sure that it addresses quite what Bison was asking (or at least what I think he was asking), nor quite what my model would analyze - the very, very short run of a single session (say, 100-400 hands). Specifically, what is the distribution of these short run outcomes? I suspect, and with no analysis or modelling yet it's only a suposition, that these outcomes may not be distributed normally.

[/ QUOTE ]
These seem like 3 different things to me too. I had an idea to do something similar to what I think you are suggesting, starting with a single hand distribution. (http://forumserver.twoplustwo.com/showflat.php?Cat=&amp;Board=probability&amp;Number=1274385 &amp;Forum=All_Forums&amp;Words=%2Bsingle%20%2Bhand&amp;Search page=0&amp;Limit=25&amp;Main=1274385&amp;Search=true&amp;where=bod ysub&amp;Name=3962&amp;daterange=1&amp;newerval=1&amp;newertype=y&amp; olderval=&amp;oldertype=&amp;bodyprev=#Post1274385) My idea was to find the most likely session result.

Clearly, you should most frequently expect to be losing early in a session, and then to climb out of the hole after some number of hands. I want to generate a graph of what this should look like. I can do this roughly assuming hands are binomial experiments but I thought it would be more cool to do some Monte Carlo simulation. /images/graemlins/cool.gif

I doubt I'll get to it any time soon though. /images/graemlins/frown.gif

Analyst
01-26-2005, 02:19 PM
[ QUOTE ]
[ QUOTE ]
Interesting stuff! However, I'm not sure that it addresses quite what Bison was asking (or at least what I think he was asking), nor quite what my model would analyze - the very, very short run of a single session (say, 100-400 hands). Specifically, what is the distribution of these short run outcomes? I suspect, and with no analysis or modelling yet it's only a suposition, that these outcomes may not be distributed normally.

[/ QUOTE ]
These seem like 3 different things to me too. I had an idea to do something similar to what I think you are suggesting, starting with a single hand distribution. (http://forumserver.twoplustwo.com/showflat.php?Cat=&amp;Board=probability&amp;Number=1274385 &amp;Forum=All_Forums&amp;Words=%2Bsingle%20%2Bhand&amp;Search page=0&amp;Limit=25&amp;Main=1274385&amp;Search=true&amp;where=bod ysub&amp;Name=3962&amp;daterange=1&amp;newerval=1&amp;newertype=y&amp; olderval=&amp;oldertype=&amp;bodyprev=#Post1274385) My idea was to find the most likely session result.

Clearly, you should most frequently expect to be losing early in a session, and then to climb out of the hole after some number of hands. I want to generate a graph of what this should look like. I can do this roughly assuming hands are binomial experiments but I thought it would be more cool to do some Monte Carlo simulation. /images/graemlins/cool.gif

I doubt I'll get to it any time soon though. /images/graemlins/frown.gif

[/ QUOTE ]

You've described something very similar to the approach I've been considering. What I want to do is come up with the single hand distribution for every one of the 169 possible starting hands. With this, I'd then run a large number (a million or so) of 100-hand (or 200, 400, etc.) sessions and empirically develop a session distribution.

The best way to get the individual hand distribution would be from real world PT databases, from long-term winning players. Outcome from each hand would be put into bins, say .2BB wide. For example, 72o would have a large spike of around .67 in the -.1BB to +.1BB bin (.67 as I mainly play 6-max; in full games, the spike would be higher as you get to fold trash a greater percentage of the time); there would be smaller spikes in the -.1BB to -.3BB bin (small blind) and -.3BB to -.5BB bin (big blind) and some small amount of other outcomes.

The problem is going to be developing these hand distributions. Do you - or anyone else reading this - know of a way to automatically extract and bin data from PT? With this, I'd be happy to run the simulation.

Analyst
01-26-2005, 02:24 PM
[ QUOTE ]
of course 100-400 hands won't be normally distributed. statistics describes a long data series... it is non-sensical to talk about statistics over 100 hands..

[/ QUOTE ]

It's not non-sensical to talk about the probability distribution of results over any size trial. Yes, "it's all one long session" but when you or I sit down at the table it's for a finite amount of time. I, and apparently others, are quite interested in the probable results.

That guy
01-26-2005, 03:44 PM
I, and apparently others, are quite interested in the probable results.

me too, this is what standard deviation is... Homer used 18BB's for 1 SD... so use your basic confidence intervals to determine the likelihood of a session...

The average results for a solid player are somewhere around +1 to +3 BB/s per hour with a Std Dev of 18...

I would love to know what normal std deviations are... I don't think 18 is that far off but who knows, it could be...

I would think Ed Miller has done some work on this given his math background...

btw, see this link for the avg BB's per hand of actual play... you have to adjust the % of times seen relative to the long-term because you can't equal weight a pair, an unpaired/unsuited hand and a suited hand:

http://www.pokerroom.com/games/evstats/totalStats.php?order=value

uuDevil
01-26-2005, 03:50 PM
[ QUOTE ]
The problem is going to be developing these hand distributions.

[/ QUOTE ]
Yeah. /images/graemlins/smirk.gif

[ QUOTE ]
Do you - or anyone else reading this - know of a way to automatically extract and bin data from PT?

[/ QUOTE ]
The structure of the Pokertracker DB is fairly easy to figure out. So I think a competant VBA programmer who is familiar with Microsoft Access should be able to fairly quickly write a macro to do this. The harder part might be to get people to share their data, though some people have done it in the past.

uuDevil
01-26-2005, 04:38 PM
[ QUOTE ]

I would love to know what normal std deviations are... I don't think 18 is that far off but who knows, it could be...

[/ QUOTE ]

Here is some data: Small Stakes SD (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&amp;Number=1362524&amp;page=&amp;view=&amp;s b=5&amp;o=&amp;vc=1)

There are other similar polls for Micro and short-handed as well.