Standard Deviation/Win Rate

[RJT]

In Mason Malmuth’s "Gambling Theory and other topics", he has a chapter on standard deviation (chapter, How Much Do You Need?). I thought I understood the concept of standard deviation. After reading this chapter, I am not sure I do.

He gives examples of having a win rate of $30 per hour and the large bankroll one needs to be assured of not going broke. I understand that these are theoretical numbers. How practical are the numbers, given his example of a $30 per hour win rate? I really don’t understand how one can have any (let alone a $30 per hour) positive win rate (redundant I know) and still know that “…it is possible to have a two-year losing streak.”

If one rolls a die 12 times, it is possible to have it come up in the following order: 1 2 3 4 5 6 6 5 4 3 2 1. That is just as likely as any other order. But how practical is knowing that?

Also, a related question - I would think that as one’s win rate (assuming based on an adequate sample size) increases one’s standard deviation would automatically narrow. Especially in limit hold ‘em. True or not true?

[jtr]

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I thought I understood the concept of standard deviation. After reading this chapter, I am not sure I do.

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Perhaps you understand it as an abstract thing, but you don't yet have a gut-level appreciation for how important this concept is and how much it will rule your life as a serious poker player.

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He gives examples of having a win rate of $30 per hour and the large bankroll one needs to be assured of not going broke. I understand that these are theoretical numbers. How practical are the numbers, given his example of a $30 per hour win rate? I really don’t understand how one can have any (let alone a $30 per hour) positive win rate (redundant I know) and still know that “…it is possible to have a two-year losing streak.”

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The numbers are absolutely practical and it is certainly possible to have an expected win-rate of $30 an hour and yet experience a 2-year losing streak. Imagine for a moment that you're playing $500 / $1000 limit hold'em. You happen to be a very marginal ($30/hr) winner in this game. Do you really doubt you could have a 2-year losing streak in these circumstances?

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Also, a related question - I would think that as one’s win rate (assuming based on an adequate sample size) increases one’s standard deviation would automatically narrow. Especially in limit hold ‘em. True or not true?

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Not true. I can see why you might intuitively feel this to be the case, but all evidence suggests that the SDs of big winners, moderate winners, and losers are quite similar. Certainly the big winner feels the bite of variance less often than the small winner, as his winnings will be growing faster and he will experience downswings less often, but his standard deviation per 100 hand session will be very similar to those of his opponents.

[dana33]

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I really don’t understand how one can have any (let alone a $30 per hour) positive win rate (redundant I know) and still know that “…it is possible to have a two-year losing streak.”

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Perhaps a concrete example will help show how this is possible.

Assume your win rate is $30/hr at 15/30 holdem, and your SD is 10 times this -- or $300/hr.

Let's say you play 500 hours per year, or 1000 hours over 2 years.

Your EXPECTED win over this time is $30 x 1000 = $20,000.

Your SD over this time is $300 x sqrt(1000) = $9487. Let's round this to 10,000 to make the calculations easier.

Statistics tells us that there is roughly a 95% chance that your ACTUAL win rate over this time is in the range 20,000 - 2 * 10,000 to 20,000 + 2 * 10,000 -- i.e., between 0 and $40,000. But note that there is a 5% chance that your win rate is greater than or less than this range. Specifically, there is a 2.5%, or 1 in 40, chance that you lose money over this time.

So, it's not LIKELY that you'll lose money over this time, but it wouldn't be a freak occurrence if it happened.

[jcmack13]

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Your EXPECTED win over this time is $30 x 1000 = $20,000.


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why exactly would 30/hr x 1000 hrs not yield $30,000?

This also changes the results of the SD calculation.

[dana33]

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why exactly would 30/hr x 1000 hrs not yield $30,000?


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Because I'm an idiot savant who can take square roots but can't multiply! (It's funny that no one noticed the error sooner.)

Correcting the error changes the specific numbers -- so that the probability of losing over two years is much reduced. (You'd need to raise the assumed SD or reduce the assumed hours to get the same 2.5% probability of losing that I erroneously calculated.) But the basic approach is still valid.

[RJT]

Thanks for both responses. They do shed light on my question. At the same time they seem to confirm what my actual thoughts were with it.

Let’s now assume that one is a winning player as defined by what many consider a good level of play. I won’t define what that is - I am sure there are variety of opinions. But, I remember reading that many consider 2-5BB/100 is good and don’t remember a number for xBB/hr. This is more what I had in mind. At similar win rates, doesn’t this then seem more unlikely to have a 2 year losing streak? Or is it still not just possible (almost anything is) but somewhat very possible.

I am not trying to belabor the point. I just want to make sure I am not missing something.

Or we can make it even easier: What win rates and Sds at relevant levels does one need to achieve to make the point moot or almost moot?

Dana 33 - thanks for your response. But if I wanted an answer like that I would have read the chapter over 10 more times. Just kidding,Dana. It does make it clearer. I don’t have a printer right now (using my laptop). But tomorrow I will print your answer and read it more carefully to make sure I understand it completely. I get the gist of it now. Thanks for taking the time to give me a concrete example.

Jtr - your response regarding SD of big winners, moderate winners and losers almost suggest that most players’ SD are about the same so long as we compare it to the same game. Are you saying this? Probably not, but can you expand on that thought a bit more for me? What effects one’s SD? Is it playing loose vs. tight, etc. Good/bad. What does one look for to better one’s SD?

[dana33]

RJT, following the pattern in my post, you can do the calculations yourself for any win/100, SD/100 and number of hands played per year that you want to assume. Then you can see how likely a two-year losing steak is in any given scenario.

Remember that Mason's article was written in the context of B&M players. If a winning player is multitabling online and playing 8 hours/day, then of course a two-year losing streak is MUCH less likely than in the scenario I presented.

[RJT]

Got it, bud. Thanks

[BugsBunny]

Jtr - your response regarding SD of big winners, moderate winners and losers almost suggest that most players’ SD are about the same so long as we compare it to the same game. Are you saying this? Probably not, but can you expand on that thought a bit more for me? What effects one’s SD? Is it playing loose vs. tight, etc. Good/bad. What does one look for to better one’s SD?

Just to address this question a little bit. Generally speaking:
Loose = higher SD
tight = lower SD
passive = lower SD
aggressive = higher SD

Following from the above a LAG will have the highest SD and a tight/passive will have the lowest SD. A TAG has competing factors at play and where his SD ends up depends on how he balances those factors.

SD is effected by [playing style/game texture] rather than earn. That fact explains why losers and winners can have similiar SD's.

As to what's the best way to lower your SD, while not impacting your earn in a negative fashion? Get better at hand reading.

[RJT]

Thanks Bugs, Buddy.