I do an exact calculation. �The math is quite similar to solving Risk of Ruin
problems, except that you calculate "risk of going broke before hitting a
royal" in place of "risk of going broke before playing forever." �One big
difference is that Risk of No Royal is always less than 1.0000 even for
unfavorable games.
I'll use F (for "Failure") for the variable that represents probability of
going broke before hitting a royal (same concept as variable R in RoR
equations). � For 9/6 JoB, F = 0.9990516541327410. � F^731 = 0.49978876,
so 731 units give slightly better than 50% chance of hitting the royal.
Probability of success (hitting a royal) is (1 - F) = 0.0009483458672590
�= 1/1054.467609892 for a bankroll of one unit. �For a bankroll of B units,
the probability of success is (1 - F^B).
The difference in our results is large enough that it should be easy to
test by simulation. �Of course, it would take a special simulation that
stops with "success" whenever a royal is hit, while stopping with "failure"
whenever the player goes broke. �I feel confidant that such a simulation
will back up my number and show that the Poisson Distribution gives a
poor approximation for this calculation (but, of course, I might be wrong).
···
On Thursday 03 November 2005 09:48 am, Dan Paymar wrote:
Steve Jacobs wrote:
>It is good to see a VP program that gives information that isn't
>all EV based.
>
>For 9/6 JoB, starting with a single unit bankroll, I compute a probability
>of 1/1054.467609892 that a player will survive to hit a royal flush.
Sorry, my method doesn't work with such small bankrolls.
>A bankroll of 731 units is needed in order to have better than a 50%
>chance of hitting a royal before going broke. These numbers are based
>on max-EV strategy.
I get 984 units for a 50% probability of a royal before going broke.
My calculations are based on the Poisson Distribution. How are you doing
it?