Hypothetical Question-Hero is runnin real bad!!

[goodguy_1]

Let's say our Hero is running really really bad.He/she continues to play tight pre-flop and well post-flop but he/she continuously is getting sucked out on in every way including the following scenarios...Multi-way pots his best hands almost never stand up and HU before the flop his hand which is the favorite rarely stands up.This goes on and on for weeks and weeks.He runs bad for 3 months and things still havent turned for Hero.


1.) What are the chances that a $3-6 LHE player, that makes $9.00 an hour after a solid but not huge 500 hr sample size,will break even over the next 500 playing hours?

I know the chances of this occurring are very very remote but I'm just curious to see what the math guys come up with here.

2.)As a variable to this problem lets say Hero's $9.00 per hr per table winrate is based on various sample sizes of 50,100,300,1000 and 2000 hours(rather than the 500 hours we use above).How do these different sample sizes effect the chaces that Hero runs this bad ?

I understand Hero could in reality not be a $9 per hour winner and that the smaller sample sizes all make that more likely with his/her poor performance.

Thnx Lee

[crockpot]

1) because money is basically a continuous variable, i'll calculate the chances of you breaking even or losing over this period.

assuming a standard deviation of 10bb/hour ($60), your standard deviation after 500 hours is 60*500^(1/2) = 1341.6. with a mean of +4500, the chances of ending up less than 0 are less than .1%.

2) i'm way too tired to answer this right now. let's just say that after n hours, your win rate will fall within this interval 95% of the time:

(measured win rate - 1.96*60/(n^1/2), measured win rate + 1.96*60/(n^1/2)

so after 50 hours, if you claim your win rate is $9, it could easily be anywhere from -$7.63/hour to $25.63/hour, and 5% of the time it won't even fall in this range!

[goodguy_1]

excellent Crock !
ty for that-very easy to understand.

What's fascinating is that the higher your hourly sd the longer total hours played you need just to be at a 95% confidence level by that I mean if you have a $60 per hr sd in $3-6 you will need 3600 hours of play just to have 60/n^1/2=1 where your hourly results would still be $9 per/hr +/- $1.96 or $10.96/$7.04 per hour.Even if you played 36,000 hours at a 95 %confidence interval you'd still have an hourly variance of $+/-0.62 cents on your hourly average of $9.Even at at n= 1 Million hours you'd still have a 12 cent variance on your hourly earn.
Just a $10 per hour increase in you hourly sd would mean that you would have to play 4900 hours an additional 1300 hours for your winrate just to be at WR +/-$1.96.

[Nottom]

[ QUOTE ]
Even at at n= 1 Million hours you'd still have a 12 cent variance on your hourly earn.

[/ QUOTE ]

And people wonder why we bash them for proclaiming themselves the best ever after posting a 5BB/hour winrate after a whole 100hours of play.

[goodguy_1]

yes indeed .I made a post 3-4 months in this forum asking for BruceZ's advice because I was freaking out that right at 1000 hours my hourly average tanked like 15% or so in 100 hours.Silly me..this puts all that in the correct perspective! 100,200,500 hours ptuey.

[bigpooch]

Yes, that's for most people that only keep track on an hour
by hour basis. But for those that keep track on an orbit
by orbit basis: ONLY need about 2 1/2 times less hands in a
ring game (even better for 6 or 5 handed) for the same kind
of confidence.

[BruceZ]

Yes, that's for most people that only keep track on an hour by hour basis. But for those that keep track on an orbit by orbit basis: ONLY need about 2 1/2 times less hands in a ring game (even better for 6 or 5 handed) for the same kind of confidence.

Ahh, that would be a nice trick if we could alter the spread of our win rate just by the way we keep track of it. Of course this is complete nonsense. The maximum likelihood estimate of your hourly rate over N hours is the total win for N hours divided by N. This is the same whether you add your results every hour, every hand, every nanosecond, or just one time in N hours. The standard deviation of this hourly rate estimate or "standard error" is SD/sqrt(N) where SD is your standard deviation for 1 hour, and N is the number of hours. How often you keep track of your results does not alter this spread. There are no shortcuts. You must play N = SD^2 hours to have 1 standard error of confidence in your hourly win rate.

Now, you can track your results every orbit and compute the standard error of your orbital win rate if you want to. If you have k orbits per hour, it will be SD/[sqrt(k)*sqrt(No)] where No is the number of orbits. Now it's true that it will take you fewer orbits to get the standard error of your orbital rate to equal 1 than it took hours to get the standard error of your hourly rate to equal 1 (got that?), but you must make the standard deviation of your orbital rate 1/k in order for the standard error of your hourly rate to equal 1, and this takes exactly the same amount of time, namely No = k*N orbits, or N hours. The standard error of your hourly RATE is k times the standard error of your orbital rate, not sqrt(k) times this as you may have thought. Remember, the standard deviation of your WIN for N hours is sqrt(N)*SD, and the standard devation of your WIN RATE per hour is SD*sqrt(N)/N = SD/sqrt(N), and the standard deviation of the WIN RATE for 1/k hours is
SD*sqrt(N)/(kN) = SD/sqrt(n) * 1/k = 1/k times the standard deviation of the win rate for 1 hour. The standard error of your hourly rate in terms of the number of orbits is SD*sqrt(k/No), so No = k*N orbits, or N hours.

For example, if SD = 10 bb, then the standard error of your hourly rate is 10/sqrt(N), so you must play N = 100 hours to make this 1 bb. If you play 4 orbits per hour, then the standard error of your rate per orbit is 5/sqrt(No), and the standard error of your rate per hour is 20/sqrt(No). To make your hourly standard error equal to 1, No = 400 orbits = 100 hours.

[bigpooch]

Right. For win rate, having more frequent statistics will
hardly reduce the size of the confidence interval for the
hourly rate; on the other hand, the interval for the SD per
hour will be estimated more quickly (but it isn't that
important as this seems to converge very quickly). It does
seem more natural to keep statistics on an orbit by orbit
basis (or perhaps every 2 or 3 orbits) because time-based
statistics in this case seems so arbitrary.

The only "shortcut" is to put in the hours and the most
common way is to play multiple tables which could (in most
cases) reduce the win rate per orbit.

[BruceZ]

on the other hand, the interval for the SD per
hour will be estimated more quickly (but it isn't that
important as this seems to converge very quickly). It does
seem more natural to keep statistics on an orbit by orbit
basis (or perhaps every 2 or 3 orbits) because time-based
statistics in this case seems so arbitrary.

The problem with computing your SD over data logged every few orbits is that your results for time periods of an hour or less for 1 table are probably not normally distributed, so if you try to estimate your standard deviation based on the sample variance of hourly results, then you can no longer use the chi-squared distribution to determine the confidence interval of the estimate, it will no longer be a maximum-likelihood estimator, and it won't be as accurate as estimates obtained from several hour sessions. Now if you *could* estimate your SD for 1 orbit or for 1 or hour accurately, and if this were a fixed distribution for each orbit or for each hour, then by the CLT you could multiply this SD by sqrt(N) for N oribts or N hours to get the correct long term SD, even though the oribit and hourly SDs are not normal.

It is best to compute your SD based on session results, where sessions are usually several hours in length, and the longer the better. Even though your hourly results are not normal, your session results will be considerably more normal by the CLT. Sampling per session has the added advantage of including in the distribution the changes due to different opponents which may change from session to session. The sessions can vary in duration, and you use the maximum likelihood estimate of the SD for variable length sessions given in the essay section. I still owe someone a derivation of that which I'll put up. After 20-30 sessions, even though the SD will still have some uncertainty associated with it, it turns out that this will have very little impact on the confidence interval of the hourly rate. After only 20 sessions, it should impact it by less than 10%, as can be seen from the t-distribution. The reason for this is that it is as likely to be estimated too high as too low, so the effect on the hourly rate confidence interval is smaller than than you might expect from the confidence interval of the SD. If you want to know the confidence interval for the SD, you could now use the chi-square distribution, since the underlying samples are normal.