Standard Deviation
 

I posted this on the internet forum, and am posting it here in hopes that someone could direct me to someplace I can find the math I need to answer my questions, or if someone has some other comments.

Basically what I want to ba able to do is find out how likely a run like this actually is.

The post:


Ok I know what everyone is thinking, another Party is rigged post. That may be true but this is different, please read the whole post before flaming me.

background:
I have been playing online poker for some time now and have long been a winning player. I have been playing mostly 100NLHE on party, and latley on poker stars.

I had a very good win rate for many months, then I noticed a sudden drop in my winrate at party poker.

Try as I might I could not find leaks to explain it, all I kept seeing was that I was no longer making much money with premium hands, I have been struggling with this for some time now, and finally found something worrysome.

I have 33,229 hands in PT since mid may (when I lost all my data). I went through the stats on the hands that have historically made me money and this is what I found:


These are based on hands where a flop was dealt, regardless of wether or not I was still in the hand, and applies to 3 FLOP cards only.

Made a pair on the flop (expected 36.7%)


Flops - Hits - %
AKx
261 - 79 - 30.2% (-6.5%)

AQx
286 - 107 - 37.4% (+0.8%)

Total - 34%, an 11% shortfall, 20% for AK.

I was willing to accept the above, but this I cannot:

Hit set on flop:
Hand - Flops - Hits - %
AA - 104 - 7 - 6.7%
KK - 98 - 9 - 9.2%
QQ - 119 - 12 - 10.1%
JJ - 109 - 7 - 6.4%
TT - 115 - 15 - 13%
99 - 146 - 11 - 7.5%
88 - 170 - 17 - 10%
77 - 139 - 15 - 10.7%
66 - 138 - 13 - 9.4%
55 - 122 - 12 - 9.8%
44 - 143 - 13 - 9.1%
33 - 134 - 20 - 14.9%
22 - 138 - 17 - 12.3%
TOTAL- 1675 - 168 - 10.029%
Expected - 204

That is a 17.6% shortfall.

One more, just for fun:

KK saw flop 93 times, 29 times an A flops (31.1%, expected 24.4%) About 33% more than axpected.


Seems as if the deck is stacked against me. How can this be explained? Please, tell me there isnt realy 52 cards in the deck, or this is actually within the realm of probability.

What realy chaps my ass is that I have still won a fair amount over these 33K hands, so I can still beat the game even with these numbers, and I don't want to quit partypoker.

33 THOUSAND hands. I am mystified. Is 33,229 hands not a rational sample size?

BTW my winrate at pokerstars is what it used to be at Party. so as Winston ChurchHill used to say, "Bugger them all!"

Ok let me have it.

-MyNameIsMud

Re: Standard Deviation
 

First of all some of you math is off. The correct way to calculate the probability of a card appearing on the flop is to calculate the probability of it not appearing, and then subtracting that value from one. In the case of flopping a set we get P=1-(48/50)(47/49)(46/48)=11.755%. Likewise for flopping and ace P=1-(46/50)(457/49)(44/48)=22.55%. These numbers are pretty close to yours anyway so I agree there is some explaining to do.

Whenever, an event either happens or doesn’t happen, you can use the binomial distribution function to determine the probability of a particular number of results in a particular number of trials. If you have Microsoft Excel, this can be done quite easily. (note that you want to use the cumulative binomial distribution function which calculates the probability of a particular number of trials or less, not exactly that number of trials).

That said, I get these results:

After 1675 pairs seeing a flop, the probability of flopping 168 or fewer sets is 1.4%

After 93 pocket KK seeing a flop, the probability of seeing 29 or more flops with an ace is 3.4%.

These results might seem suspicious but you need to keep a few things in mind. First some perspective. The probability of rolling a 12 while playing craps is 2.8%. You have seen results around as likely as rolling a 12. Second, developing a hypothesis (that Party’s deal is not fair) after looking at the data does not constitute proof. Hundreds of statistical anomalies are possible in the data. For example, you could flop flush draws when holding a suited hand or straight draws with a connecting hand at improper frequencies. That a couple of statistical anomalies are found is expected.

If you truly believe the flops are not fair develop your hypothesis and then collect the data. Keep track of your next 93 flops with KK and your next 1675 flops with pocket pairs. If these results repeat then you might be on to something.

Paul

Re: Standard Deviation
 

[ QUOTE ]
After 1675 pairs seeing a flop, the probability of flopping 168 or fewer sets is 1.4%

[/ QUOTE ]

So if 200 players were to play this many hands of holdem over the same period, at the same on-line poker site, and if the poker site IS fair, you'd expect 3 of these players to have these results, right?

And another 3 with the opposite results (they are just happier people [img]/images/graemlins/smirk.gif[/img])

Re: Standard Deviation
 

[ QUOTE ]
[ QUOTE ]
After 1675 pairs seeing a flop, the probability of flopping 168 or fewer sets is 1.4%

[/ QUOTE ]

So if 200 players were to play this many hands of holdem over the same period, at the same on-line poker site, and if the poker site IS fair, you'd expect 3 of these players to have these results, right?

And another 3 with the opposite results (they are just happier people [img]/images/graemlins/smirk.gif[/img])

[/ QUOTE ]

Three (2.8) would be your average number if you repeated the 200 players playing 1675 flops many times. You would get exactly three only about 22% of the time.

Paul

Re: Standard Deviation
 

Thanks for taking the time to do this and not calling me a nut.

-MyNameIsMud

Re: Standard Deviation
 

[ QUOTE ]

These results might seem suspicious but you need to keep a few things in mind. First some perspective. The probability of rolling a 12 while playing craps is 2.8%. You have seen results around as likely as rolling a 12. Second, developing a hypothesis (that Party’s deal is not fair) after looking at the data does not constitute proof.


[/ QUOTE ]

Actually I deleloped the hypothesis about 25K hands ago, (that is, a shorage of flopped sets, and a shortage of flopped pairs with AK and AQ (my primary non-pair raising hands)), but I can not prove that.


[ QUOTE ]

If you truly believe the flops are not fair develop your hypothesis and then collect the data. Keep track of your next 93 flops with KK and your next 1675 flops with pocket pairs. If these results repeat then you might be on to something.


[/ QUOTE ]

This is underway now. I will complete this and post the final results, whatever they may be, as well as make available my PT database and all the emailed hand histories from party, which covers 40K hands now.

playing 3-4 tables, I am up to about 7000 hands now, and the results have thus far are a bit worse than those of the first 33K.

Flopped sets are down to 19/221 (8.6%)
Pair or better on flop AK, AQ 25/146 (21%)

I know you all think I am nuts, but I intend to prove this one way or the other, and, as this trend is driving me mad, I need to vent a bit while I grind out these hands. I will also be seeking help from the psychology forum.

-MyNameIsMud

Re: Standard Deviation
 

You should only expect to make a pair (or two pair, trips, or quads) with an unpaired hand on the flop 32.43% of the time, so you are actually getting luckier than average here.
In addition, if you always raise with these hands preflop, you will see somewhat less than this because when people play against you it is somehat more likely that they will have some of your aces, kings, and queens.

33K hands is a rational sample size, but it still will have reasonable expected deviation.

Craig

Re: Standard Deviation
 

[ QUOTE ]
You should only expect to make a pair (or two pair, trips, or quads) with an unpaired hand on the flop 32.43% of the time

[/ QUOTE ]

Can you show me how you come up with 32.43?

-MyNameIsMud

Re: Standard Deviation
 

This is 1-P(the 6 cards that match yours do not show up on the flop):

1-(44/50)*(43/49)*(42/48)=0.32428

or about 32.43%

Bozeman would not steer you wrong on this kind of problem.

Re: Standard Deviation
 

[ QUOTE ]
In addition, if you always raise with these hands preflop, you will see somewhat less than this because when people play against you it is somehat more likely that they will have some of your aces, kings, and queens.


[/ QUOTE ]

Yes, that is exactly what I was going to say. Good point. These are not showdown simulations. Some times you won't even see a flop when you have premium cards. On the actual flops seen, you should expect a lower number of premium cards. Unless there is a flop 100% of the time, then you cannot logically expect there to be as many aces, kings, and queens.

Don