looshle, on Wednesday, July 30th, 2008, 2:36 PM, said:
You obviously know a hell of a lot more math than I do but I can't see how variance will GROW over time. Variance is bigger the smaller the sample size, so if you are making 0 EV bets you are going to mathmatically break even and should even be closer to even the more hands you play amirite?
Yes and no. :)The problem is that if you are playing the same game N times, there are two definitions of variance:1) The variance per game
. This variance decreases
proportionally with the number N of games.2) The total
variance. This increases
proportionally with the number N of games.The really important concept, by the way, is the standard deviation, which is the square root of the variance, and which states how much you can expect to be "away from average". To give an example in numbers:Suppose i play a game with a standard deviation of $1. This means that after a single game, I will usualy be between one dollar down and one dollar up from my expected result. Now if I play this game 100 times, I get a standard deviation of $10. (Multiply the original number by the square root of 100.) This means that I will usually be between -$10 and +$10 away from my expected result. Per game
this means I will usually be away less then $10/100 = $0.10 from my expected result.Thus, we see that even though the total
standard deviation grows (from $1 for one game to $10 for 100 games), my standard deviation per game
has decreased (from $1 for one game to $0.10 per game for 100 games).Now the nice thing of having a positive EV is that in the end, EV will outrun standard deviation. Say that for this game, I have an EV of $0.25 per game. Then after playing one game, the standard deviation per game is four times what I expect to win, so very often, I will lose instead of win. On the other hand, after 100 games, the standard deviation per game
is smaller than my EV per game, so for the vast majority of the time, I will be up.Thus, after 100 games, my expected "swing" from average will be about $10, but I will expect to have made $25, so no matter the direction of the swing, I will most likely still be up. Thus, the total "swing" increases, it just gets outrun by the EV because that one increases even more. Or vice versa, the total swing per game
decreases until it gets below the EV.Now if we have an EV of exactly zero, it will of course never outrun the swing. That is, my swing per game
still decreases, but the thing you should really be interested in is your total
swing in this case, since there is no EV that will in the end compensate for your swings. Thus, you will drift away from zero more and more - the good news is just that the chance of drifting away upwards is as large as the chance of drifting away downwards. :)Well, those were a lot of words... I'm not sure if they clarified anything, but I hope they did.