Steve Jacobs wrote:
[A simulation] can certainly be used to determine which of two significantly different
answers are more likely to be correct. However, there isn't enough
simulation power on the planet to confirm/deny the 15th decimal
place of an exact calculation. In addition, simulation breaks down
when extreme accuracy is required, because random number generators
are never perfect. So, technically, simulation can never be quite as
good as an exact calculation.
You're right. I worded that badly. A simulation can't prove whether or not some other method is exact, but if the results are quite different it gives a strong suggestion that one method or the other is wrong.
I don't think anyone here is interested in the 15th decimal place. Three to five significant figures are more than adequate for anything we do, and a G5 Macintosh certainly has enough number-crunching power to achieve 5-digit accuracy with a Monte Carlo simulation in just a few minutes. As another writer to this forum said, the 3 to 5 royal bankroll rule of thumb is adequate for many players.
Here's my equation again:
F = p(lose) + p(1)*F + p(2)*F^2 + ... + p(50)*F^50
Look familiar? F is the probability of starting with a one unit bankroll
and going bust without ever hitting a royal. Note the lack of a term
for the payoff of a royal. That's what makes this a "risk of no royal"
equation rather than a "risk of not playing forever" equation. The
risk of no royal depends only on the other payoffs.
Sorry, I guess I didn't relate it to the form of the Sorokin formula because of the different variable names, plus it wasn't expressed using a Sigma symbol for the summation. Also, the fact that the Sorokin equation won't converge if the total ER is less than 1. Does this equation really converge naturally, or do you have some special technique to get it to converge?
This whole discussion took a different turn than I had anticipated.
I haven't seen your article on computing probability of royals,
since I don't subscribe. I figured you had probably followed the
same path that I took, using a Sorokin-like equation, and the most
likely outcome would be that my numbers would match yours. I've
been working with "best-shot at royal" strategies for over a year.
Perhaps you would like to write an article on the subject for Video Poker Times. The payment would be a complimentary one-year subscription.
Similar equations can be used for other payoffs. For example, to find
the probability of eventually hitting 4/kind, you can start with the
Risk of Ruin equation and remove the terms for the 4/kind payoff.
For 9/6 JoB, this leads to F = 0.97773237412 when using max-EV
strategy. This gives a 1/44.9 chance of hitting 4/kind when starting
with a single unit bankroll. To have a 50% chance of hitting 4/kind,
one needs a bankroll of 31 units. A strategy that is optimized for
surviving until hitting 4/kind can improve this to 1/43.7720, reducing
the 50/50 bankroll to 30 units.>Sorokin doesn't really deserve credit for this, since the same method
>was used by Laplace, De Moivre, Lagrange and Bernoulli, hundreds
>of years ago.Then why did not someone come up with a single formula for games with
a wide range of probabilities and payoffs before Sorokin?If you haven't seen it elsewhere, perhaps you've looked in the wrong
places. Gambling literature is the wrong place to look. Try math books
that discuss Markok chains and/or random walks.
Obviously you've more up on math subjects than I am. Sorry to say, I've never heard of Markok. (Hmmm, sounds like a Klingon name, doesn't it? LoL) I had a course on LaPlace transforms in college, but never applied them to anything. Also had a course on analog computers, which of course became useless as digital computers got faster. Newton-Raphson (sp?) iterations I've used a lot, though.
> They may, or they may not back anyone's theoretical calculations.
True. It is possible that we are both wrong. It isn't possible that we
are both right, since there is a big gap between our answers. The
important thing isn't really who is right and who is wrong, the important
thing is to find the truth and move forward with a better understanding
of how these things work.
I noted that my method was not good for small bankrolls because it is geared to the royal cycle length. OpVP restricts it to bankrolls at least adequate for one cycle without a royal, but even that may be too small. I feel quite confident that my use of the Poisson Distribution gives a good approximation for large bankrolls, but most players who want a Risk-Before-Royal calculation are probably interested in small bankrolls, so your method is probably better.
I believe that if you'll give some serious thought to using a Sorokin-like
equation, you'll realize that it is the correct approach and that it gives
an exact solution.
I agree, at least in theory, and I intend to work more on that after I get the simulation working so that I have some reasonable approximations for checking the results.
Thanks for your input,
Dan
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Dan Paymar
Author of best selling book, "Video Poker - Optimum Play"
Editor/Publisher of VP newsletter "Video Poker Times"
Developer of VP analysis/trainer software "Optimum Video Poker"
Visit my web site at www.OptimumPlay.com
"Chance favors the prepared mind." -- Louis Pasteur