Variance and Poker


"A measure of the dispersion of a set of data points around their mean value. It is a mathematical expectation of the average squared deviations from the mean." Answers.com

Basically what this means is that say you expect a certain result (x) to be the average or mean result and (y) is the outer bounds(positive or negative) of the possible results. y in this situation would be our variance. An example that relates to poker could be say we expect to win $20 per session at a cash game. Some times we will lose $40 and then again sometimes we will win $80 but over a period of time we expect to win $20. Another simple example could be a hand where you expect to win 80% of the time. In this same situation spread out over say a 1000 sample you should win 800 times and lose 200 times but there is no guarantee that you will win 800 times. It may take 10,000 hands for it to even out.

Lets look at a simple example situation where you are 80/20 and lets assume that the results will equal out over a 100 hands. Lets say you are betting $4 at a time and when you win you will double up and when you lose you will lose the $4. Using a standard $EV calculation our expected value would look like this over 100 hands (.80*8) - (.20*4) * 100 = 560. Another way of looking at it is 80 times we win 8 (640) and 20 times we lose 4(80). Now we subtract out loses from our wins 640 - 80 = 560. So, we could expect to profit total of $160(560-400) over 100 hands. However, this will not always be the case.

Now lets look at another example situation with the same as above(betting $4 on an 80/20) except this time the odds equal out over a 1000 hand sample. Lets break it down into blocks of 100 where they do not always come out 80/20 but do over the entire 1000 hand sample. In the end we should be +1,600. Lets say that the blocks of 100 hands are as follows:


Variance over 1k sample
win lose profit
90 10 +280
100 0 +400
40 60 -320
60 40 -80
80 20 +160
70 30 +40
95 5 +340
90 10 +280
85 15 +220
90 10 +280
800 200 +1,600

As you can see from the example table above just because we are 80/20 we do not always come out +160 ahead in our 10 100x hand blocks. Our highest point is +400 and our lowest is -320 but over 1000x hands we equal out to +1,600 as we should. That was a simple example showing one situation over a period of time where we are are 5:1 favorite. We could potentially lose 200 hands in a row but still end up ahead over 1000 hands.