Jean-Baptiste Queru wrote:
If 1000 players each play 1 hand of VP, you'll have 1000 measurements.
Let's assume JoB for a second.
Quite a few players will hit nothing, i.e. they "measure" 0. Some will
hit a high pair, i.e. they "measure" 1. Some will hit two pairs, i.e.
they "measure" 2. And so on. Put all the numbers together, use your
usual formula, and you get the variance for that set of samples.In VP however, we know precisely what the theoretical distribution is
supposed to be, so we can actually compute the theoretical variance of
the game - it's the variance that you get when each possible deal is
considered the same number of time, and where for each deal each
possible draw is considered the same number of times.
I think that would result in a huge variance value that would be totally unrealistic since no player tries all 32 ways of playing every hand. That's what a game analysis does, but then it selects only the draw with the highest EV (i.e., it assumes "perfect" play). After examining all 2,598,960 possible dealt hands this way, it can compute the probability of each final hand (again assuming perfect play). The game variance is then calculated from those probabilities, and the standard deviation is just the square root of the variance.
However, you need to be careful: when your students do their
measurements, you'll probably get a distribution that's fairly
"normal" (i.e. a bell curve). In VP however the distribution is
absolutely not a bell curve, i.e. the usual measurements done on a
bell curve don't apply to a single sample (e.g. you can't say that 16%
of players will be more than 1 standard deviation below the average).
Actually, for the case of 1000 players each playing one hand, I can say that exactly 0% (none) of those players can be more than one SD below expectation. That is because the standard deviation on one play is 4.418 unit bets, and each player is risking only one unit bet. I can also say that 2.516% of the players can be expected to be more than one SD above expectation since that is the percentage of final hands that return more than 5.418 unit bets.
But note that this is far different from the case of one player playing 1000 hands. For 1000 perfectly played hands, I calculate the SD to be 144.27 unit bets. With a little more math, we could calculate the probability of being more than one SD ahead or behind at that point.
A nice property of statistics is that the average of "enough" samples
of any distribution is close to a bell curve (that's why the normal
distribution has been studied so much). That means that if you get
your 1000 players to each play 10000 hands and consider their final
result, you'll have something that looks more like a bell curve. How
much is "enough" is a fairly open to interpretation. I personally
assume that 10 million samples are likely to be "enough" for VP,
though I have not studied that issue in detail. (I will, when I find
some free time during which I decide to not practice DB).
How many hands is "enough" is subjective, and I don't think it is important. The Risk of Ruin for a given game and bankroll is much more significant, which is why my program, Optimum Video Poker, offers the Sorokin formula calculation as well as plotting SD curves.
Dan
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Dan Paymar
Author of best selling book, "Video Poker - Optimum Play"
Editor/Publisher of VP newsletter "Video Poker Times"
Developer of VP analysis/trainer software "Optimum Video Poker"
Visit my web site at www.OptimumPlay.com
"Chance favors the prepared mind." -- Louis Pasteur