RORBR goal

[Dan2]

Steve Jacobs wrote:

There has been a lot of discussion about the possibility of playing
forever and never going bust or hitting a royal, even though the ER
sans RF is less than one.

Jonathan has argued that the probability of this happening is
non-zero. I agree that the probability is non-zero, but it is so
small that it's insignificant. For all practical purposes, it can be
assumed to be zero, so I agree with Steve that it can be ignored.

Just to clarify, my position isn't that this probability is so small that
it can be ignored, my position is that this probability approaches
zero as the number of hands played approaches infinity, and that
it is mathematically impossible to play forever without hitting a royal,
no matter how favorable the non-royal payoffs are made.

Then we'll have to agree to disagree on this point. The probability
of playing forever is non-zero. Your statement that it approaches
zero is an acknowledgement of that fact. Thus is it theoretically
possible for it to happen.

I have shown that "approaching infinity" in our case is about
100,000,000 hands, or perhaps 4,000 or 5,000 sessions, for a lifetime
pro. For most of us, 1,000 sessions is probably approaching infinity.
Going beyond that has no practical value.

Therefore, the probability is so small that for all practical
purposes it can be ignored.

Dan

···

On Thursday 17 November 2005 09:22 am, Dan Paymar wrote:

--
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

[Non-text portions of this message have been removed]

[Steve_Jacobs]

Your interpretation of my statement is, umm, rather odd.

Let's see what math says. Let p be the probability of hitting a royal
on the very next play, while q = 1-p is the probability of any other
outcome. For a player who is allowed to play forever (perhaps by
having unlimited credit), the overall probability of hitting a royal is
given by:

p(royal) = p + p*q + p*q^2 + ... p*q^n + ....

where the term p*q^n represents hitting a royal after failing to do
so on the first n attempts. This reduces to:

p(royal) = p * (1 + q + q^2 + q^3 + ... + q^n + ...)

Whenever q is less than one, the series in parentheses converges
to 1 / (1 - q), so we have

p(royal) = p / (1 - q) = p / (1 - [1 - p]) = p / (1 - 1 + p) = p / p = 1

so the overall probability of a royal is EXACTLY 100%. Note that
this result is independent of any or all payoff value(s) so it makes
no difference whether the player faces a favorable or unfavorable
game during the stretches when no royals occur.

When the player is required to stop playing after losing his/her bankroll,
then the only time that the player fails to continue indefinitely is when
the player does, in fact, go broke. The remaining cases, which do
continue indefinitely, must eventually hit a royal.

Therefore, for any starting bankroll, the only two possibilities are
that the player will go broke or the player will continue indefinitely,
and those players who continue indefinitely are certain to eventually
hit a royal. Thus, my formula for RoRBR is exactly correct, for any
VP game.

I've explained this to Dan in private email, but he was not convinced,
and offered no credible counter argument. If he wishes to disagree with
the math, there is little I can do to disabuse him of his false notions.

···

On Thursday 17 November 2005 10:55 am, Dan Paymar wrote:

Steve Jacobs wrote:
>On Thursday 17 November 2005 09:22 am, Dan Paymar wrote:
>> There has been a lot of discussion about the possibility of playing
>> forever and never going bust or hitting a royal, even though the ER
>> sans RF is less than one.
>>
>> Jonathan has argued that the probability of this happening is
>> non-zero. I agree that the probability is non-zero, but it is so
>> small that it's insignificant. For all practical purposes, it can be
>> assumed to be zero, so I agree with Steve that it can be ignored.
>
>Just to clarify, my position isn't that this probability is so small that
>it can be ignored, my position is that this probability approaches
>zero as the number of hands played approaches infinity, and that
>it is mathematically impossible to play forever without hitting a royal,
>no matter how favorable the non-royal payoffs are made.

Then we'll have to agree to disagree on this point. The probability
of playing forever is non-zero. Your statement that it approaches
zero is an acknowledgement of that fact. Thus is it theoretically
possible for it to happen.