Something to mention. Is the minimum bankroll for the 5% risk of ruin correct? That wound correspond to a 90 percent confidence interval. Yet above they show 70 percent and 95 percent confidence intervals which makes me wonder if they’re confusing the two tails of a 95% interval with the one tail of a 90 percent interval. Also, how are they computing this number or does it come from a simulation?

Mason

I don't know anything about this, but... another site has a risk of ruin calculator which gives the same answer.

That site links to this wikipedia article:

Thanks,

Mason

I hope this simple summary can help you guys understand "variance" a bit better.


3 most important variables
========================
- Sample size = # hands (in CG at least) - bigger sample = higher degree of confidence, all else the same;


- WR (bb/100) = speaks for itself - higher your WR, the greater confidence you can have over what your WR over that sample is, all else the same. This is perhaps one of the most important variable because it affects your experience of 'variance' disproportionately to the increase in WR - every incremental increase by x amount in WR will decrease the effect or experience of variance more than the last incremental increase of x amount.

- "std dev bb/100" (you can find it under this label in HM3 + PT4) = Measures how volatile your playstyle is; function of playstyle. More volatile you are - the bigger pots you play on average; more frequent getting all in etc., the higher your std dev bb/100, the bigger the effect of variance (magnitude of swings, duration of swings - within short sessions as well as in the long run)

Winrate not effect variance. Only sample size and std dev is important when you describe variance. Winrate only define which zone you will hit the variance.

When std dev and sample size is the same 10bb winner and 2bb winner will experience same variance. The only different part 2bb winner spend more time on loosing side.

Since variance is calculated by comparing each sample to the mean (i.e., winrate), if the mean of a dataset changes, then in general the variance of that dataset will change as well.

Let's say you somehow have full clairvoyance on your opponent's range and strategy. Let's also say you're risk averse, maybe you're playing on a 10 BI roll and can't redeposit, so you pass any marginal bluffcatching spot that happens, e.g. you never call a B50 by the river if you have less than 30% equity. This HAS to reduce your variance, because these are spots you're expected to lose often to then win big just often enough so you make a profit. This also reduces your winrate because you're passing +EV spots.

I think there should be spots where the relationship inverts itself (maybe someone else could come with an example) but there is a relationship between winrate and variance.

A bigger sample size does not produce a higher degree of confidence. What it does do is to allow for the confidence interval around the estimate to be smaller. For example, you can always form a 95 percent confidence interval no matter the sample size. It’s just that the width of the interval will be smaller with a larger sample.

Mason

You’re correct. The relationship between winrate and variance/standard deviation is important, especially for medium lengths of time. For detailed discussion, see my gambling theory book:

https://www.amazon.com/gp/aw/d/18806...=UTF8&qid=&sr=

Mason

I still cant understand how winrate effect variance.

V = [1(n-1)]*sum[(x[k] -<x>)^2] where n is number of samples, x[k] is the kth sample and <x> is the sample mean, given by

<x> = (1/n)*sum(x[k])

Winrate is one of the parameters used to calculate variance. If winrate changes, so does variance.

The win rate itself doesn’t affect variance, it’s how you achieve that win rate that affects variance. In poker there are high risk plays that can increase your win rate a little but can drive your variance up more than a little. Again, see my gambling theory book for more discussion.

Mason

Winrate not effect variance its Standart Deviation that effect winrate.

How you achive that also mean Standart Deviation.

I'm not completely sure what you mean but there is a statistical formula for the variance, and the square root of that formula is the standard deviation. Also, in my expanded gambling theory book there is a chapter titled "Increasing Expectation at the Price of Variance" and chapter titled "But This Doesn’t Always Happen" which addresses ways you can increase your win rate and lower variance.

Mason