Size matters (math question)

[burningyen]

I scoured the board for a thread like this but didn't find one even though this topic comes up again and again.

Let's say you decide to dedicate the rest of your life to playing $109 buy-in, 100-player MTTs, same prize structure (35%/21%/14%/10%/6%/4%/3%/3%/2%/2%), same opponents, and neither you nor your opponents ever get any better. How many of these MTTs would you have to play before you would know your ROI to within:

+/-100%
+/-50%
+/-10%
+/-5%

I know I'm displaying near total ignorance of statistics here, but I'm hoping one of you guys can do this sort of thing. I'm just trying to figure out to what extent we can know our ROI in our lifetime.

[Exitonly]

this is what i'm working into my next update for my spreadsheet... i've just been too lazy to crack open my stats book, and get it done. maybe i'll take a shot at it now.

[Wanabe]

Exit could you please show a link to your spreadsheet,let me know if i have to donet money ...
thanks in advance.

[Jurollo]

Exit,
If you eventually need web space I can throw it on my site as one of a few places to host it.
~Justin

[Exitonly]

yea i remembered you offered, i was going to PM you when i finished with this stuff

--

Spreadsheet

Heres the link to it ... no donations needed.

[EverettKings]

Post this in the probability forum and you'll get some really sharp cats who breathe this stuff to do it for you without breaking a sweat.

It's basically a problem of rolling a wighted 100-sided die and using variance to construct confidence intervals, stuff that I did once upon a time but would take significant effort to relearn. Somone there probalby knows it off the top of their head.

Everett

[pzhon]

[ QUOTE ]
$109 buy-in, 100-player MTTs, same prize structure (35%/21%/14%/10%/6%/4%/3%/3%/2%/2%), same opponents, and neither you nor your opponents ever get any better

[/ QUOTE ]
Let's suppose you finish in every place with equal probability. Then the standard deviation would be $440, or 4.04 buy-ins.

The standard deviation of your observed ROI after n tournaments would be 404%/sqrt(n). If you want to have a 95% confidence interval of +- 10%, you need your standard deviation to be 5%, so n=(404/5)^2=6,531 tournaments.

That is more than most individuals play, but I'm sure that as a group, 2+2 forum regulars have more data than that. So, I suggested that we pool our data, but that post received no replies, and sank like a stone.

If you are a winning player, your distribution of places is not uniform. This will tend to increase the standard deviation. If you make k times as many final tables, with an even distribution at the final table, your standard deviation will increase by almost a factor of sqrt(k). for k=2, the standard deviation would be $606, or 5.57 buy-ins.

[Exitonly]

man.. i used to think i was the [censored] at math.


i gotta start reading more statistics stuff, that sounded interesting. Just wish i had a good idea of what it meant.

[pzhon]

[ QUOTE ]
Just wish i had a good idea of what it meant.

[/ QUOTE ]
Here is an application:

Your bankroll should be at least c*SD^2/ROI, where SD is standard deviation and c is a number that depends on your risk tolerance and ability/willingness to move down if you hit a bad streak. Most people seem happy with a value of c between 2 and 4.

If your ROI is 40%=0.4 buy-ins with a SD of 5 buy-ins, then your bankroll should be at least c*5^2/0.4 = c*62.5. For c=2, that is 125 buy-ins. For c=4, that is 250 buy-ins.

For comparison, the SD for 1-table SNGs is about 1.7 buy-ins. In LHE, it is about 15 BB/100 for online full games, or 17 B/100 for 6-max. In NLHE, it is about 30-70 BB/100.

If you know your win rates in each game, you can use these to determine what gives you the best hourly rate for your bankroll.

[burningyen]

Thanks for the detailed response, pzhon. Something about your post strikes me as counterintuitive, though, or maybe I'm just not understanding what you mean by SD. I thought SD is essentially what laymen refer to as "margin of error," which I always thought of as being totally dependent on sample size and not results. But you mention that someone with a non-uniform distribution will have a higher SD. If that's the case, I would have thought that the more bunched the outcomes, the smaller the SD. If someone makes the final table every time in his first 100 tournaments, are you saying that we will need a bigger sample size to figure out his ROI than for someone with a uniform distribution?