Has anyone come up with a ROR formula that includes...
 

Has anyone come up with an ROR formula that includes the effect of not knowing your "true" win rate?

What I mean is, pretty much everyone is just guessing their true win rate. Those with 1 million hands in Poker Tracker obviously can make a much more accurate guess than those with 20K hands, but it is still a guess.

Now, I realize there are formulas both for ROR calculation based on bankroll and WR, and formulas for determining the level of confidence for your WR based on your current WR and the number of hands played.

I am wondering if anyone has combined the two. Meaning, the formula has independant variables of the hands you have played, the WR for those hands, and your bankroll size. The ROR from this new formula would usually be higher, unless your current WR is negative in which case this formula will not necessarily give a 100% ROR (since, there's a chance you might actually be a winning player).

Maybe this has been done before and I haven't heard about it. I am sure it can be done though, and would likely give a more accurate figure than simply guessing what your true win rate is and plugging it into the ROR formula.

Re: Has anyone come up with a ROR formula that includes...
 

A man named William Gossett did this about a century ago. He wrote under the name of "Student" (he worked for Guiness Beer and they didn't want the notoriety of having a statistician in their breweries), so the result is called the "Student's t" distribution.

However, the correction is not very large if you have played a large number of hands and aren't asking questions about much larger samples of future hands. While uncertainty about your win rate is important, there are much more important factors like nonconstant win rate and rare events that make the computation less accurate than theoretical bounds suggest.

Re: Has anyone come up with a ROR formula that includes...
 

Actually, the t-distibution is for the case where the standard deviation is unknown.

Re: Has anyone come up with a ROR formula that includes...
 

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Has anyone come up with an ROR formula that includes the effect of not knowing your "true" win rate?
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Now, I realize there are formulas both for ROR calculation based on bankroll and WR, and formulas for determining the level of confidence for your WR based on your current WR and the number of hands played. I am wondering if anyone has combined the two.

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The concept isn't too hard, though the execution may be. Just integrate, over all possible win rates, the ROR formula multiplied by the normal density based on your sample.

If it turns out to yield a readable formula, this would be very interesting. Nice question!

Re: Has anyone come up with a ROR formula that includes...
 

Bill & Jerrod's MOP will feature this. Coupla more months, sigh...