A quick note on stats and convergence

[OrigamiSensei]

My mini-rant is based on a hand post that came out in the last week that I can no longer find but that I felt the need to comment on. In the hand post the heads-up stats were provided for a player with 46 or so hands of data, during which time they had a PFR stat of either 2% or 4%, can't remember which. At some point I remember reading at least one reply that basically assumed the PFR number was accurate based on this very small number of trials. Now for people versed in statistics and the long run this is obviously a large leap of faith, but for people new to the game and new to tools such as PAHUD and GameTime+, particularly micros players this may not be so obvious. So if the experienced micro'ers and the folks who understand math and statistics will bear with me on this one I think it's worth reminding newbs to be careful about trusting these numbers overmuch with small sample sizes.

In the thread I can no longer find the oppnent being analyzed had raised either once or twice in the space of around 46 hands, I don't remember the exact numbers. But for our purposes it will be sufficient to examine the possibility that this seeming rock is actually a normal TAGgy 9% PFR type on a cold run of cards. By going back to my old probability textbook I was able to track down the formula for the binomial distribution. This allows us to calculate the probability that a particular event will occur n times in y trials, with p representing the probability of that particular event:

f(y) = (n select y) p^y*(1-p)^(n-y) where (n select y) is n!/(y!*(n-y)!)

If we fill in the numbers where 9% (p=.09) is the actual pre-flop raise probability for this player, n is a span of 50 hands and y is either 1 or 2 depending on whether the player raised once or twice (thus looking like a 2% or 4% PFR player) we get the following:

A 9% PFR player has a 10.7% probability he will raise only twice in a given 50 hand span and a 4.4% probability he will raise only once. Adding the two together is over 15%, or close to 1 in 6 times.

Now, are we ready to really assume that this player only raises 2% or 4% of the time absent any other evidence? We shouldn't be quite ready to yet. Now, of course if we see the guy open-limp and show down hands like AJ from the cutoff then that's pretty strong evidence the heads-up display is accurate. But absent other evidence, particularly if you've been multi-tabling and not taking mental note of things like that you can't rely so completely on the number.

Like I said, obvious to the more experienced but hopefully it's a thought of some use to newer players. Sorry I lack the mathy knowledge to fully calculate confidence intervals and such but I didn't want to turn this into a thread for SMP or the Probability forum, just wanted to throw out something to consider.