Game Volatility at Baccarat
| 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.
