


PART
III
The purpose of this article is to explain in a probabilistic sense the degree
of volatility that can be expected.
By Andrew MacDonald
Executive General Manager, Gaming Risk Management, Crown Ltd 1998 
Introduction 
Mathematics 
Variable Betting 
Conclusion 
A question often asked by Analysts and Senior Management within the casino industry
is: "… when will we hit our theoretical win percentage?"
This question is generally reserved for when results are below expectation and
pressure is being exerted for the results to "turn" in the casinos favour.
Some labour under the impression results must "turn" so that the average
will be achieved over time. "A run of ill fortune means that it is "our turn"
and we are more likely to win." Such thoughts should obviously be reserved for
those on the other side of the gaming tables. There is no "evening up". The game
has no memory, with each result being effectively independent. What will occur
as the number of decisions is increased is that the "expected" deviation from
the mean in percentage terms will decrease. In absolute terms, however, the deviation
will increase as decisions increase. This can best be shown as follows:
n

sqrt (n)

sqrt (n/n)

TW (A x n)

100

10

10%

1.25

1000

31.6

3.16%

12.5

10,000

100

1.0%

125

1,000,000

1000

0.1%

12,000

Where n equals the number of decisions and the square root of n represents one
standard deviation from the expected result and where the expected result equals
1.25% x n for the game of Baccarat.
An interesting further point is that if we were to conduct an experiment where
an unbiased coin were to be thrown 100 times with the results of each throw recorded
and the experiment temporarily stopped after the first 10 throws with 10 heads
having been thrown, it would now of course be more likely for heads to still be
ahead at the conclusion of the 100 trials. Remember, the coin has no memory so
for the 90 throws we would expect a 45/45 split with a standard deviation from
the mean being the square root of the number of decisions multiplied by the probability
of winning multiplied by the probability of losing.
P (H) 
=0.5 X 90 
=45 

1SD 
=sqrt (npq) 
=sqrt (90 x 0.5 x 0.5) 
=4.74 
95% of results will fall within +/ 1.96 standard deviations of the mean.
Therefore, for the next 90 throws we would expect the following:
45 hands +/ 1.96 x 4.74
ie. 95% of results will fall between 35.70 and 54.30
Add this to the first 10 heads and it becomes obvious that at the conclusion of
the 100 coin tosses it is highly likely heads will still outweigh tails if the
first 10 results were all heads. No evening up, in fact the ability only to project
that if we start out behind we are more likely to end up behind and if we start
out ahead we are more likely to end up ahead.
What is useful to understand for casinos conducting high end play is how many
decisions would be required for a certain departure from theoretical to occur
for a given confidence interval.






