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Mathematics of Blackjack Discussion
Re: ROR CALCULATION?
Posted By: lucky In Response To: Re: ROR CALCULATION? (Don Schlesinger)
Date: Friday, 3 March 2006, at 5:33 p.m.
Below is an article that I found with a risk formula.... My question is this how is the asw calculated is it just the average bet squared based on a large amount of simulated bj hands? Or is it variance + wph^2? He used 3.887 as the asw in the example... Was the ave wager then the sqrt(3.887)? Any help is appreciated!
I. What is the risk of ruin for a given bankroll and given win goal?
This can be estimated using the standard ruin formula,
with the bankroll adjusted to reflect the standard deviation, as
shown by Griffin in "Theory of Blackjack". The average squared
win is very nearly the variance (i.e. the standard deviation
squared), more precisely it is the variance plus the square
of the win per hand. You can approximate the game of blackjack
by betting the square root of the average squared wager
on a biased coin with P(heads) = 0.5 + wph/2sqrt(asw), where asw is
the average squared wager and wph is the expected win. So,
here P(heads)= 0.5+.0151/(2sqrt(3.887)) = 0.5038. The ruin formula is:R = (1 - S^b)/(1 - S^(a+b))
where R is the probability of ruin from 0 (none) to 1 (always)
a is a'/sqrt(asw), the coin toss bankroll
b is b'/sqrt(asw), the amount to be won in coin toss units
S is P/(1-P), the ratio of a coin winning to losing
P is 0.5 + wph/2asw, the bias of the coin
wph is the win per hand
asw is s^2 + wph^2, the average squared win
Suppose we wish to double a bankroll of 300 basic units. The bet size
effectively reduces this to a=b=300/sqrt(3.887)=152.16 units.
S=0.5038/(1-0.5038)= 1.0153. So R=(1-1.0153^152.16)/(1-1.0153^(152.16*2))
=9.0%.