All righty Fellas...Stupid Probablity Question

[etgryphon]

Alrighty then...

First off never taken a real probablity class or have a good probability book (incidentally any suggestions for a good book?). I'm pretty good with math (Physics and CompSci double Major) so you don't have to put it in kiddie pool terms.

So this is what I'm trying to figure out. I'm trying to find a confidence interval for what someones preflop raise and VP$P based off of the number of hands that you have played with them.

For example: If you have played 50 hands with a Villian who has a 5% PFR what is your confidence that they do in fact have 5% PFR and they are not just on a good/bad streak for them.

5% of hands is basically 99+, AK\AQ, Some AJ in LP or other random raises.

This seems like a pretty easy problem to figure out because it is based off of discrete random events (i.e. there are only 1326 hands that they could have PF)

So how do I figure it out?

-Gryph

[LINES]

Good question, give me a bit to figure this out.

[eOXevious]

I think you also need to know how often they fold after the flop to get a good confidence rateing. I'm good at figuring out weird probabilities, but I still don't quite understand what you are asking.

[etgryphon]

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I think you also need to know how often they fold after the flop to get a good confidence rateing. I'm good at figuring out weird probabilities, but I still don't quite understand what you are asking.

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Wow maybe this wasn't as easy as I thought... Alright let me try to explain it better.

Lets take the Extreme case:

You have played an infinite amount of hands with the Villan. Your stats so that the Villian has a PRF tendency of 5%. So based on the infinite hands that you have play with him and normalized distribution then you know that he raises 5% of the 1326 total possible Hold'em hands:

C(52,2) = 1326
x = # hands that Villian raises PF.

x = 5% * 1326 = ~67 hands (which is roughly, 99-AA, AK, AQ)

I want to figure a confidence interval (or maybe its standard diviation? Remember...no statistics training)

Were I can say:

I have played X number of hands with Villian with PFR of p % +- SD

For example:

For 50 hands, I am confident that Villian plays 5% of his hands +- 2%

For 100 hands, I am confident that the Villian plays 5% of his hands +- 1%

etc.

Hope this is clearer...

-Gryph

[Siegmund]

If ALL you know is that someone has played N hands against you and VPIP'ed or raised some percentage of them, you have a simple binomial distribution.

The standard error of your estimated p is sqrt(p*(1-p)/N); for instance, if someone puts money in on 20 pots out of 50, you have an estimated VPIP of 40% with SE of 7%, and your 95%CI would be 26-54%: in other words, in 50 hands, statistics alone will only spot the tightest rocks and loosest fish. It's even harder with rarer events. (5 raises out of 100 could easily be a 2%PFR player running hot or 10%PFR running cold.)

In the long run these stats can turn out to be useful. In the short run, LOOK AT THE CARDS. If someone raises UTG and shows down 77 or A8s, you have learned a LOT more from that one deal than you can from 100 hands worth of statistics.

[Delphin]

I have been thinking about this a lot lately. One problem is that every player really has at least 10 different VPIP's for a full ring game (one for each position). There aren't just 1326 possible events. There are at least 1326*10 possible events (receiving each possible hand in each possible position).

And even this doesn't factor in the action prior to the player's turn. Obviously my VPIP on the button is much different if there there is a raise, reraise, 3 bet, and cap, than if it is folded to me. The space of all possible preflop situations is quite large.

When you have a 100 hand sample, you only have 10 samples in each position at best. If the player was running hotter or colder in one or two positions than normal (which is overwhelmingly likely), then you have a large potential for over/underestimating the player's true VPIP. If there is a maniac at the table raising every hand, or if the player is seated to the left of several LPP players this will have a large effect on the observed VPIP.

There is a general consensus that VPIP converges very quickly. I think that I believe this relative to other stats (BB/100 in particular). I think that an estimate obtained over 50/100/200 hands aren't nearly as reliable as most people think.

I would love to see a detailed mathematical treatment of this problem. I can easily do the calculation for one position assuming that other players previous action has no effect. Once I try combining multiple positions or accounting for other players actions I get lost.

[Delphin]

[ QUOTE ]
The standard error of your estimated p is sqrt(p*(1-p)/N); for instance, if someone puts money in on 20 pots out of 50, you have an estimated VPIP of 40% with SE of 7%, and your 95%CI would be 26-54%: in other words, in 50 hands, statistics alone will only spot the tightest rocks and loosest fish. It's even harder with rarer events. (5 raises out of 100 could easily be a 2%PFR player running hot or 10%PFR running cold.)


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If you have 100 hands at a full table, that is 10 hands in each position. Assume you observed 30% VPIP in late position, 20% VPIP in middle position and 10% VPIP in early position, with 40% VPIP in SB and 0% VPIP in BB. Using the binomial distribution error formula gives the following error estimates for each position:

B 30% +/- 14.5%
CO 30% +/- 14.5%
MP3 20% +/- 12.6%
MP2 20% +/- 12.6%
MP1 20% +/- 12.6%
EP3 10% +/- 9.5%
EP2 10% +/- 9.5%
UTG 10% +/- 9.5%
BB 0% +/- 0%
SB 40% +/- 15.5%

Total 19% +/- ???

This gives me an estimate of 19% VPIP for the player, but I have no idea how to combine those error estimates. If I combine the VPIPs first and then compute the error using the formula I get 4% error, but if I average the error in each position I get 11% error. Which (if either) is more valid? /images/graemlins/confused.gif

[atnels]

[ QUOTE ]
If there is a maniac at the table raising every hand, or if the player is seated to the left of several LPP players this will have a large effect on the observed VPIP.

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Just came across this thread, and since I'm bored at work on a Friday afternoon I'm going to start thinking about coding this up. What you mention here is the most intriguing part about the problem. I believe the whole thing will become a function of some type of "Table Aggression Factor," since a decent # of hands like AQo are obviously easy open-raises from anywhere on the table but could be easy folds in other spots when some mouse raises UTG and you're UTG+1. Other times, some uber-lag raises UTG and another donk reraises, you -could-be capping with AQo....

How is this going to affect the numbers of a good TAG over 100k hands? Probably not too much. But, it will have a drastic contribution toward the numbers over ~100 hands or so, which seems to be the motivation for asking this question in the first place....

I'm trying to figure out some kind of weighting to allow you to normalize and compare an EP raise in a weak/tight game with a LP cap in a loose/passive game.

Interesting post.