8 SNG set probabilities

[AleoMagus]

Well, this likely only interests me, but I'll post it here anyways in case someone else cares.

I calculated probabilities for, and graphed every unique $ finish possible in a set of 8 SNGs (as this is my daily regiment these days)

This is actually pretty cool, because it is an exact combinatoric calculation, and gives a confidence result, but without relying on Normal distribution assumptions.

an excel file called 'sng set probabilities' can be found at:

http://www.aleomagus.freeservers.com/spreadsheet

You can fiddle around with the input values at the top so that it refects your individual finish breakdown. It is currently set for $22 SNGs, (I've been 4-tabling those twice a day). If you change it for another buy in, you should adjust the scales on the charts to accommodate min and max values for your particular buy-in. Hopefully you'll see what I mean.

I think it I find some time, I am going to do this for an even larger set of SNG outcomes. 20 would be about the most I'd ever want to do on excel because the combinatorics get crazily huge after a while. Still, even with a 20 SNG set, it would give a much clearer picture of how the actual SNG results curve compares with standard normal distribution estimates. Actually, even this little 8 SNG sample does that to some extent.

What I really should do is learn to program and write something that would do much larger samles automatically. It really would not be out of the question for a fast computer to do these same kinds of calculations for a 100 or even 1000 SNG sample. In this way it would be possible to give EXACT confidence figures for even very large samples and predict confidence values exactly for future results over an upcoming sample that may also be very large.

Anyways, hopefully this interests someone
Any questions/thoughts?

Regards
Brad S

[ThorGoT]

Wow. This is way cool. I admit I don't fully understand it, but it is still way cool.

[AleoMagus]

Just in case anybody is having trouble figuring it all out, I'll offer a bit more explanation.

In the first graph, each point represents a possible $ outcome of a 8 SNG set. This is represented by a $ amount on the x-axis (you profit or loss) and a probability value on the y-axis.

In the 2nd and 3rd graphs, each point still represents a $ amount as an outcome, but the probability values are now the probability of that result AND every result better (or worse, depending on the graph).

It is possible to see then your odds of 'making the money' after an 8-SNG set, and other things like your odds of making more than $100 etc...

What is most interesting about this is that unlike my confidence calculator, this does not rely on normal standard distribution assumptions. This is simply a combinatorics calculation based on the respective probabilities of every possible outcome.

AND, where this gets really interesting is in the potential to expand this for a much larger and more useful SNG set.

It effectively gives confidence values that are not approximations at all.

Regards
Brad S
For example

[poboys]

I could write a perl script that takes the following variables Pr(1st), Pr(2nd), Pr(3rd), the %-Juice (eg. 0.10 for $55 and below SnGs), and the # of SnGs.

The result would be a .csv file that you could import and graph in Excel (or OpenOffice). If you are really interrested, PM me and we can talk about the specifics.

[Daliman]

[ QUOTE ]
I could write a perl script that takes the following variables Pr(1st), Pr(2nd), Pr(3rd), the %-Juice (eg. 0.10 for $55 and below SnGs), and the # of SnGs.

The result would be a .csv file that you could import and graph in Excel (or OpenOffice). If you are really interrested, PM me and we can talk about the specifics.

[/ QUOTE ]

Bah on both of you. Find a program that tells you RAISE! /images/graemlins/laugh.gif

J/K, pretty cool, if obtuse to laymen such as I, stuff.

[tallstack]

Brad,

I have been playing around with your exact combinatoric calculation idea in VB and I have a crude program that will spit out the probability and winnings data pairs for sets up to about 160 games. Anything more than that and the combination values are out of range for the double precision variable types. I am not a programmer so maybe there is another variable that will let it go higher. Either way, this should be high enough to show convergence with any normal distribution model.

I don't have any webhosting, so if you are interested then send me a PM and I will forward it to you. If you like it then feel free to post it.

Dave S

[Irieguy]

Dude, thank you. I just started 8-tabling, so it's nice for me to be able to have a firm grasp on the frequencies of my respective session outcomes.

I would love to see the same for bigger numbers. Multiples of 8, of course, would be ideal since you also play 8-game sessions.

Thank you very much.

Irieguy

[eastbay]

I just wanted to say: very nice work, Aleo.

eastbay

[stupidsucker]

Thanks much.

[AleoMagus]

Ok, Dave (tallstack) did some excellent work and sent it to me and I have put it up on that site also.

It is the 'Setsof 8to50SNGs(1)' file (or something like that)

As it's title states, his file contains the same graphs of 8,12,20 and 50 SNG sets. It's pretty interesting. I cannot believe how smooth the 50 SNG set curve is. I know that people have been saying the normal distribution calculations are a good estimate for a while now, but I just didn't really believe it for some reason. I am starting to now. At least between the 10-90% range, it looks spot on. Those are my first glance assessments anyways.

The next step is graphing the same normal distribution results and comparing them. Who knows. We may be in for a surprise yet.

Anyways. Great work Dave.

Regards
Brad S

[AleoMagus]

Wow. Check this out

http://www.aleomagus.freeservers.com/spreadsheet

the 'confidence comparison.xls' file

Regards
Brad S

[Irieguy]

Wow. I love it when I believe something for a long time and then find out that I'm flat wrong.

A normal distribution. Who woulda thunk it?

Irieguy

[w_raedy99]

It sounds like you are doing these things in Excel so I am not sure if Excel supports anything more precise than double-precision floating point.

If you were doing it in Java, you could use a solution called BigDecimal. Note: I am a Java programmer and if you could show me the calculations, I could very likely write a program to do it as well. Java even has packages that would allow you to produce the graphs as well.

There are other solutions too. Depending on how good of a programmer you are, you can always do bit manipulation to use two double-preicision floating point numbers to act as one or a similar idea.

Hopefully this will be helpful,
Will

[tallstack]

Hi Will,

Thanks for the info. I used Visual Basic for this, and then exported the won/lost and probability data into Excel for graphing. VB is the the only language that I know anything about, and I am definitely not a strong programmer. If you would like to see the code then just send me a PM and I will forward it to you. It is only about 100 lines of code so you would abe able to get it working in Java in no time I am sure.

It the end, it looks like the results for SNG sets higher than about 50 games smooth out to look normally distributed anyways. So there may not be any need to go higher.

If you know of a variable type with a higher range in VB, then please let me know and I will change the code. Basically, it would need the capacity to store a number about 10^2500 to get to 1000 games.

Dave S

[eastbay]

[ QUOTE ]
Wow. Check this out

http://www.aleomagus.freeservers.com/spreadsheet

the 'confidence comparison.xls' file

Regards
Brad S

[/ QUOTE ]

Absolutely beautiful work, Aleo. The result with the evidence should be recorded for all time in the FAQ.

eastbay

[tallstack]

Very nice!

The more I think about this, the more surprised I am that it fits so nicely after just 50 games. It seems to be about the only thing that does converge after 50 games /images/graemlins/laugh.gif

Great work, Brad!

Dave S

[stupidsucker]

could someone explain the 'confidence comparison.xls' file to me? I understood the other file, but this one I didnt get.

[AleoMagus]

Any past calculations that some of us have done regarding confidence estimates (like my confidence calculator in the SNG spreadsheet) have all been based on Normal distribution assumptions.

This has generally been considered a good estimate, but the idea has been tossed around a few times that maybe SNG results do not obey a normal distribution. The discrete nature of SNG results (only a few distinct possible outcomes) seemed to indicate that in some ways it might not make sense and the actual distribution might be quite different.

Anyways, the combinatoric calculations here are not estimates at all, but are exact results based on summing all possible outcomes from a small sample of SNGs (like 8-50).

The comparison essentially shows us that while the Normal distribution is still an approximation, it is, in fact, a VERY good approximation. The two graphs look almost identical. They are so alike that I do not intend to do any further combinatoric calculations for larger SNG samples. The usual Normal distributuion calcualtions are much simpler and apparently give us virtually identical results. In a strict mathematical sense, they are not equivalent, but they are so close it is unimportant for our purposes.

This has been surprising to me. Call me a sceptic, but I just didn't expect it to match up nearly this well. This is a pleasing result however, as it takes a lot less work to do these kinds of calculations with the usual stats.

Just to illustrate:

a 1000 SNG sample has 4^1000 distinct outcomes (roughly speaking, that's a 1 followed by 600 zeroes). There are of course tricks to make this easier to compute, but it is still very calculation intensive. Using the normal distribution calcualtions is as easy for a 1000 SNG sample as it is for a 10 or 1000000 Sample.

Regards
Brad S