Prob. Test Question

[Richard Tanner]

I don't post on this forum usually (in fact this is my first time) so please forgive me if this has already been mentioned here. That said I got into a lively debate with someone in an online sit-and-go last night.
We were arguing the merits of testing probablity and sample size. More specifically, he was attempting to say that Pokerstars software is designed to pay off the underdog in pre-flop all-ins 75% of the time. To support this he showed two test groups of 200 pre-flop all-ins that showed this percentage.
I said that I believed his sample group to be to small, and that to eliminate varience (i.e. there is no rule that prohibts 100 coin flips from landing heads, it's just unlikely) he need a much larger test group.
I brought up theorys like "Gambler's Ruin" and the like but he insisted that "the Fix was in" and that this proved somthing was amiss.
Did I miss something or is this wrong.
Thank you for any replies

Cody

[Mike Haven]

he's right

all sites have incredibly complex software that weeds out the potential losers and gives them good hands to make them win

they don't want only a million dollars a day in honest rake - they want more

[jimymat]

unfortunately he's right

[Precision1C]

200 test is too small a sample set. In a Malmuth book he calculates that if you are in the 99th percentile unlucky you can be at a loss after 2 years of full time play with an expected value of one big bet per hour. Though I suspect that this isn't the largest issue here. It is likely you are the victim of "schooling" combined with survivorship bias in the testing. In micro and low limit Texas Hold'em people call pre-flop with all kinds of junk that can make big hands such as random suited cards and with unsuited connector hands. As long as 9 or ten people see the flop and play loose after the flop this strategy isn't that bad. For instance there are 9 small bets in the pot with 9 active players, as long as you expect 5 others to call it pays to go for a 4 outer hand since you are getting 14:1 on your bet plus a net gain on later rounds if you do hit your straight or flush. If the pot was raised pre-flop and there were lots of callers the pot odds on the flop become huge and even a back door flush might be enough to call the flop bet. This is the "schooling" phenomenon, where all the fish are in the pot making it right for all of them to call with marginal draws since so many bets are going into the pot. If there was only one fish in your game he will hemorrage money but since there are 9 of them you will hemorrage money pressing hard with top pair into this many draws. Survivorship bias in introduced by looking only at the hands that call on the river to figure out how unlucky you are. For instance you raise pre-flop with AK unsuited catch As 6h 4h and get 8 callers on the flop betting round people will be getting in excess of 17:1 so calling with any hope at all looks right. For instance 6s 9s can call on the 2 outer trips, backdoor flush draw and the small chance that low two pair could win. The key here is that no legitimate hand should fold here so you are up against all kinds of draws, low pair, middle pair, gutshot and 8way straight draws and flush draws. On the turn many of the backdoor straights and flushes that didn't improve will fold bringing the pot down to 3-5 people. On the river likely the hand will come down to 2 or 3 people and the person with top pair good kicker will lose a huge pot to two pair, trips or a draw that came home. This looks highly unlucky but really isn't because instead of racing versus 1 or 2 people he was trying to survive 9 people drawing against him. As an exercise you can deal yourself AA and 9 other random hands and see how many hands you win on the river. I suspect that it will be on the order of 20-25% if all the hands go to the river. One way to think this situation is that your top pair good kicker beat 8 of the nine people who called pre-flop but lost to the surviving drawing hand that beat you.

[pzhon]

[ QUOTE ]
More specifically, he was attempting to say that Pokerstars software is designed to pay off the underdog in pre-flop all-ins 75% of the time. To support this he showed two test groups of 200 pre-flop all-ins that showed this percentage.

[/ QUOTE ]
If the 200 were chosen in an unbiased fashion, then 75% wins for the underdog would be overwhelmingly strong evidence that the PokerStars deals are not random. Don't believe anyone who tells you the sample size was too small. That's a huge difference from what is expected. It is over 7 standard deviations away from the mean.

If you see 200 hands, and the underdog won 75% of the time, this is overwhelmingly strong evidence that either the deals are not random, or that the hands were not chosen in an unbiased fashion.

If you mail me a coin you think is fair, I may send you a videotape showing 100 heads in 100 tosses. What are the odds of that happening? I'll conveniently delete the times the coin turned up tails. This proves almost nothing about the coin. <font color="white">So, why bother? I'll just keep the coin. Thanks.</font>

Someone proposed a pattern in the Ultimate Bet deals. He said that it had happened something like 62 out of 65 times, when par was less than 50%. Exactly how often it was supposed to occur was unclear, but when I tested it (including data from his table), the pattern happened about 15 times out of 35. My conclusion was that he had selective memory, and simply ignored the times the pattern missed.

[Jman28]

I second everything phzon said, but I wouldn't steal your quarter. That's wrong.

Also, it's impossible to prove anything by any amount of statistical data if by 'prove' you mean 'be absolutely certain.'

The question really is how convinced do you need to be.

The numbers just show you how probable it is for a fair deal to have those results.

But yeah, read phzon's post again. And check the part he bolded if you haven't already.

[fnord_too]

Quick bit of math. If you flip a fair coin 200 times, the likelhood that it will come up heads 50 times is, I believe:

C(200, 50) * 0.5^200 =~ 2.82*10^-13. (twice this if you say 50 Heads OR 50 Tails)

In contrast, getting the expected 100 heads you have:
C(200,100) * 0.5 ^ 200 =~ 0.056

The chance of any specific sequence of outcomes occurring is 0.5^200. You cannot take a given string of 200 outcomes and say "The chances of that happening were 0.5^200, ergo there's something funny going on!" because some string has to occur. The thing with getting exactly 50 heads is that the number of possible strings with exactly 50 heads is very small in relation to the total number of possible strings.

I don't know the standard devation of heads over 200 flips, but I do know the mean is 100, and the chances of getting 50 or less is pretty damn small.

Let's say by underdog he meant something so close to 50/50 that it doesn't matter. Lets also say that by 75% he meant between 70% and 80%. The likelyhood of this happening is still very very small, I'm not going to plug through the math, but I'm confident the number is less than 10^-8, and that is being pretty generous.

Now a lot of questions arise:
1. Does he in fact have 200 cases documented or is that his mental running tally?
2. Was he correct about who the underdog was in each case (89s IS a favorite over 33, though many incorectly believe that the 33 is a slight favorite).
3. (As already asked) Was the sample randomly taken?

I don't know the answer to any of these questions, but I do know that the experiment is easily reproducable, so why not reproduce it and see if you get the same results.

One other note: he said that his data "proved" something. In science, experiments do not prove theories, they only support or fail to support theories. In math you prove things, but you do not run experiments nor rely on data to do so.

I don't know that I contributed anyover what pzhon said, but I did like remembering how to compute random walk result probabilities.