Setting Table Maximum Betting Limits
| Risk and volume dependence in relation to determining maximum betting limits for independent trial, negative expectation table games. By Andrew MacDonald Senior Executive Casino Operations, Adelaide Casino, 1995 |
Casino Analyser Reference Maximum Bet Volatility |
Introduction | Key Principles | Setting Limits | Mathematics | Effective Maximum Bet Limits | Variable Betting | Volume Dependence | Maximum Loss Point | Variable Bet Distribution | Non High-End Casino Operation | Analysis of Results | Conclusion |
| Win percentage Game variance Variable expense percentages Fixed expenses Taxes, fees etc. As an example consider the following game with equal-sized bets :- |
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| Game : | Baccarat | ||||||||||||||||||
| Win percentage : | 1.25% (ref 3) | ||||||||||||||||||
| Variance : | 0.97% (ref 4) | ||||||||||||||||||
| Variable expenses : | 0.7% of turnover (indicative example only) | ||||||||||||||||||
| Fixed expenses : | $500,000 per annum (example only) | ||||||||||||||||||
| Tax : | 13.75% of win (example only) | ||||||||||||||||||
| Fees : | 13.5% of after tax and expense profit. (example only) | ||||||||||||||||||
| The expected return of 1.25% is based on a mix where approximately 60% of the bets are placed on Banker and 40% of the bets are on Player. It is assumed that N hands of equal bet size are played. The standard deviation of a sample is equal to the standard deviation of the population (square root of the population variance) divided by the square root of the sample size. Therefore the standard deviation of the expected percentage return is: | |||||||||||||||||||
| = v (97%) / sqrt N | |||||||||||||||||||
|
= 98.5% / sqrt N
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| The expected percentage return represents a random variable which is normally distributed with mean 1.25 and standard deviation 98.5 / sqrt N. Then the quantity Z where | |||||||||||||||||||
| Z = ( y – 1.25 ) / s.d. | |||||||||||||||||||
| has approximately the standard normal distribution ie. the probabilities for the Z values are approximately equal to the areas under the standard normal curve. Therefore the probability that the return (Y) is negative ie. that the player wins after N equal-bet size games is | |||||||||||||||||||
| Prob( Y < 0 ) | = Prob ( Z < ( 0 – 1.25) / s.d.) | ||||||||||||||||||
| = Prob ( Z < – 1.25 sqrt N / 98.5) | |||||||||||||||||||
| Therefore the unfavourable end of the 90% probability interval is obtained by solving for the percentage return y where -1.645 represents the Z-score or number of standard deviations that y is to the left of the mean | |||||||||||||||||||
| ( y – 1.25) sqrt N) / 98.5 = – 1.645 (ref 5) | |||||||||||||||||||
| Thus for N hands | |||||||||||||||||||
| Y = N * 0.0125 – N *1.645 * 0.985 / sqrt N | |||||||||||||||||||
| For example if N = 1000 then -38.7 units is the lower limit on loss in terms of a uniform bet size.
Prior to fixed expenses and fees, the following table is produced for the lower end of a 90% probability interval (1 in 20 chance of the displayed result or worse occurring). Table 1 |
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|
Number of Hands
N |
90% lower
probability limit |
Win net of tax
& commission |
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|
Win
|
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|
1000
|
12.5
|
-38.7
|
-40.4
|
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|
5500
|
68.8
|
-51.4
|
-82.8
|
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|
10500
|
131.3
|
-30.0
|
-106.4
|
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|
15500
|
193.8
|
-7.9
|
-115.4
|
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|
20500
|
256.3
|
24.3
|
-122.5
|
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|
25500
|
318.8
|
60.1
|
-126.7
|
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|
50500
|
631.3
|
267.2
|
-123.0
|
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|
100500
|
1256.3
|
742.7
|
-62.3
|
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|
150500
|
1881.3
|
1252.8
|
27.0
|
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| The maximum loss point occurs at: | |||||||||||||||||||
|
34500
|
431.3
|
130.4
|
-129.1
|
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| ______________________________________________________________ ref 3 relative edge at the game of Baccarat based on relative probabilities of 0.5068 (bank) and 0.4932 (player) and a 60:40 mix of bets. ref 4 average sum of squares for the game of Baccarat at a 60:40 mix of Bank and Player bets. ref 5 normal distribution table value for a standard deviation that provides a 5% tail at either end of the distribution. ______________________________________________________________ |
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