Setting Table Maximum Betting Limits

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Firstly, the maximum loss point, of the left tail of a probability interval for win net of tax (and commission if applicable), can be determined by establishing the relevant number of hands that will minimise the loss. The loss is minimised at the turning point in the curve which can be established by differentiating the function of the curve. The only variable in the equation for determining win net of tax is the number of hands. Therefore, by differentiating with respect to the number of hands, it is possible to calculate that number of hands which will minimise the loss.

Let :

Number of Hands = N
Tax = T%
Commission = C%
Win percentage = E%
Variance of a single game = V
Probability limit = Z (1.645 for 90% limit, 1.96 for 95% limit etc.)
Profit net of tax and commission = P

Then :

P = ( E%.N-Z. sqrt (N.V)).(1-T%)-N.C%
P = N.(E%.(1-T%)-C%)-Z.(1-T%).sqrt (N.V)
dP/dN = E%.(1-T%)-C%-1/2.Z.(1-T%).sqrt (V/N)
The turning point occurs where the differential equals zero.
0 = E%.(1-T%)-C%-1/2.Z.(1-T%).sqrt (V/N)
Z/2.(1-T%)sqrt(V/N) = E%.(1-T%)-C%
sqrt(N) = {Z.(1-T%)/(2.(E%.(1-T%)-C%))}.sqrt(V)
N = ((Z.(1-T%))/(2.(E%.(1-T%)-C%)))^2.V

As an example the following is provided:-

Then :

N = ((Z.(1-T%))/(2.(E%.(1-T%)-C%)))^2V.
N = ((1.645.(1-13.75%))/(2.(1.25%.(1-13.75%)-0.7%)))^2.0.97.
N = 34,136.
and,
P = N.(E%.(1-T%)-C%)-Z.(1-T%).sqrt (N.V)
P = 34,136.(1.25%.(1-13.75%)-0.7%)-1.645(1-13.75%).sqrt (34,136.0.97)
P = -129.1

This compares to the result shown in Table 1 on Page 5 of :

The maximum loss point occurs at :

Calculation in the above manner represents a more efficient method than tabling a series of results and establishing the maximum loss point from that listing. As the number of hands required is generally relatively low this is a technique which is more readily applicable in high limit junket operations. Otherwise the establishment of the forecast number of hands (decisions) in the minimum period under consideration should over-ride this where that result is greater than the number of hands at the maximum loss point.