Variance Calculator

[LCposter]

I apologize if this has been posted before, but I've searched and not found these calculations posted to date. After running breakeven over 400 games, I wanted to determine the likelihood of this occurrence. I realize that this can be esimated using Monte Carlo simulation, but I found that unsatisfactory knowing that a precise answer can be obtained.

What follows is a general-purpose series of calculations that will determine likelihood of a given hot/cold run for any site, any buy-in, and even any payout structure. Someone with a more extensive math background may want to double-check my work, but I believe this methodology is correct.

In order to keep the calculations simpler, the model assumes that the finish distribution for the prize places is uniform (i.e. for top 3 paid, p(1) = p(2) = p(3) = ITM/3). There is no assumption about the finish distribution for OOTM places. It certainly would be possible to expand the calculations to allow for different frequencies for 1st, 2nd, and 3rd if there was interest.

Inputs:
Let B1 = buy-in
Let B2 = rake
Let B3 = # of players per tourney
Let B4 = # of places paid
Let B5 = ITM rate (0 to 1)
Let B6 = length of run (# of games)
Let B7 = profit for given run

Outputs (calculations written in Excel form):
Set E1 (expected return for loss) = -B1 - B2
Set E2 (expected return for win) = (B3 * B1) / B4 + E1
Set E3 (expected profit per game) = B5 * E2 + (1 - B5) * E1
Set E4 (expected ROI) = E3 / -E1
Set E5 (standard deviation per game) = SQRT(B5 * E2 ^ 2 + (1 - B5) * E1 ^ 2 - E3 ^ 2)
Set E6 (expected profit for run) = E3 * B6
Set E7 (standard deviation for run) = E5 * SQRT(B6)
Set E8 (z-score) = (B7 - E6) / E7
Set E9 (prob of running equal to or worse than given run) = NORMSDIST(E8)

I've of course put this into an Excel spreadsheet, but I don't know how to share that on the forum. If anyone does, let me know. Otherwise, if you create your own spreadsheet, you can double-check your work by checking against the following sample data set:

ITM of 39%, 400 games, breakeven, PokerStars $25+2:
B1 = 25, B2 = 2, B3 = 9, B4 = 3, B5 = 0.39, B6 = 400, B7 = 0

Calculated results:
E1 = -27. E2 = 48, E3 = 2.25, E4 = 0.083333, E5 = 36.581, E6 = 900, E7 = 731.625, E8 = -1.2301, E9 = 0.1093

I think some of you should find this useful. Personally, I found it therapeutic to see that about 11% of the time, someone with a similar ITM rate will run worse than I have. Hopefully, you don't find out your run is in the bottom .001% (if so, maybe you're being too generous with your ITM% estimate).

One note of caution: The probability is calculated based on the assumption that the profits for the length of run follow a normal distribution. This is certainly a safe assumption for long runs (say n > 100) but not a great assumption for short runs (say n < 10). So you'll get more accurate answers for longer runs. Enjoy!

[Misfire]

I think someone on here at one time had a program that spit all this out real nice and neat.

[lacky]

eh, i think i'll just play and bitch my way through the hard times. Figuring out that mess of numbers and letters would be far more deppresing and frustrating than running bad.

[LCposter]

I thought about making a program but thought people could be wary about installing and running an executable from an unknown source. Anyway, I thought I might as well point out one other application of these formulas, since the subject often comes up on the forum.

"Am I am winning player?" This question comes up frequently, and the stock response is "play a few thousand SNGs and if you are still up, you probably are a winner player." The mathematical answer to this question is not that difficult to obtain. Working with the formulas from the original post, add one more:

Set E10 (breakeven ITM) = E1 / (E1 - E2)

For Party Poker (11s, 22s, and 55s), the formula will calculate E10 as 0.33 (which should come as no surprise because you need an extra .03 ITM to cover a 10% rake).

So plug this value into the B5 cell (ITM rate), and your # of games and profit into B6 and B7, and E9 tells you the confidence with which you can determine if you are a winning poker player.

Here are some examples if you play Party 11s and have a ROI of 10%:
Up $100 over 100 games, P(winner) = .738, 26% chance you've just been lucky
Up $250 over 250 games, P(winner) = .843, 16% chance you've just been lucky
Up $500 over 500 games, P(winner) = .923, 8% chance you've just been lucky
Up $750 over 750 games, P(winner) = .960, 4% chance you've just been lucky
Up $1000 over 1000 games, P(winner) = .978, 2% chance you've just been lucky

Statistically, 95% confidence is usually the threshold for saying the data supports the claim, so if you have a 10% ROI, you need about 750 games to say you're a winner. Of course, it takes fewer games if your ROI is higher and more games if you ROI is lower.

[rvg72]

http://www.rvgsoftware.com/ROISimulator.zip

Let's you plug all that stuff in and simulate as many samples as you like.

You need .NET Framework 1.1

rvg