## Wednesday, September 08, 2004

### StatKing

As are many other bloggers, I too am participating in a Fantasy Football League. I've been the stat-keeper for this league for about 10 years now, and during this time have gone by the moniker of StatKing. We'll statistics and probability can prove useful in poker too and that's where this post comes in.

The question: Is there something up w/Party Poker and Q,8 showing up on the flop more often than it should?

As Pauly mentioned in his blog, the other night it did seem that Q,8 was showing up every 5th hand or so. In order to determine if it was showing up with abnormal frequency, we'll have to do some calculations. Basically, how often does a flop of Q,8,x show up? The answer should go something like this:

There are 6 combinations of flops where there is only 1 Q and 1 8. They are:

Q,8,x
Q,x,8
x,Q,8
x,8,Q
8,Q,x
8,x,Q

Each of the above flops is equally as probable as the next, so we only have to calculate the probability of one of those flops occurring, then multiply that by 6.

So, the probability of the first combination is the quantity: (4/52)*(4/51)*(44/50) = .0053092 or just over 0.5%. Multiply that by 6 and you get .0318552 or about 3.2%. So a flop of Q,8, rag should show up about once every 31 hands or so.

I then grabbed my latest 100 hand history from Party Poker and did a little string searching. The PokerProfessor should be able to appreciate this command:

grep -i flop hand_history.txt wc -l

My bash shell returned 92, so there were 92 flops seen in that 100 hand history file I downloaded. The next command was to determine how many of those flops had a Q and 8 in them.

grep -i flop hand_history.txt grep Q grep 8

The result returned 6 flops that matched that query. However, one of them was Q,8,8 so I discarded that. Out of 92 hands, 5 had that "magical" Q,8,x flop or 5.43% of the time.

Is that extraordinary? You can't really answer that question without knowing the variance associated with that event occuring. In statistics, events that either occur or don't, with no in-between, fall into what's known as the binomial distribution. The variance of that distribution is calculated with the following formula:

Variance = n*p*(1-p) where n is the number of trials and p is the probability of success.

The standard deviation is simply the square-root of the variance, so the standard deviation of the success of that flop is (92*.0318552*(1-.0318552))^0.5 or 1.6844.

In the 92 trials, the expected number of flops that should show Q,8,x is n*p or 92*.0318552, 2.93.

To summarize, I should have seen a flop like that about 3 times, when in fact I saw it 5 times. However, my limited sample size shows that it occurred at a rate only 1.2 standard deviations greater than that average, which is hardly that abnormal. Therefore, I cannot conclude that anything is "up" with the flops at Party Poker.

To truly be sure, I'd need several thousand sample sizes, but I'm convinced that wouldn't tell me anything is out of the ordinary anyway, so I'll end my analysis here. Hope you had fun :)