#### Calculation of Area Under the Standard Normal Curve

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, 04-25-2012 at 04:53 AM (11135 Views)
On occasion I will post code to the blog as an extension of the Sim Secrets chapter of my book. These posts will generally only be of interest to programmers.

Page 91 inTheory of Blackjackand page 147 inBlackjack Attackhave tables used to determine areas under sections of the standard normal curve. These tables can be used to determine the probability of some events. For example,CVCXuses this to determine the probability of your actual results in a casino.

The formula for calculating the area to the right of a point (x) on the curve is 1 minus:

This is the cumulative distribution function (image pieced together fromWP). In Excel, it is calculated using Norm.Dist. But, it’s a royal pain to estimate in code. So, I have included the code I use below, taken from a long forgotten French site years back and adapted for the problem at hand:

Function normp(z As Double) As Double

Dim zabs As Double, q As Double, pdf As Double, p As Double, expntl As Double

‘Normal distribution probabilities accurate to 1.e-15.

‘Z = no. of standard deviations from the mean.

‘P, Q = probabilities to the left & right of Z. P + Q = 1.

‘PDF = the probability density.

‘Based upon algorithm 5666 for the error function, from:

‘Hart, J.F. et al, ‘Computer Approximations’, Wiley 1968

Const p0 = 220.206867912376

Const p1 = 221.213596169931

Const p2 = 112.079291497871

Const p3 = 33.912866078383

Const p4 = 6.37396220353165

Const p5 = 0.700383064443688

Const p6 = 3.52624965998911E-02

Const q0 = 440.413735824752

Const q1 = 793.826512519948

Const q2 = 637.333633378831

Const q3 = 296.564248779674

Const q4 = 86.7807322029461

Const q5 = 16.064177579207

Const q6 = 1.75566716318264

Const q7 = 8.83883476483185E-02

Const cutoff = 7.071

Const root2pi = 2.506628274631

GoSub normpcalculs

Exit Function

normpcalculs:

zabs = Abs(z)

‘***** |Z| > 37.

If (zabs > 37#) Then

pdf = 0#

If (z > 0#) Then

p = 1#

normp = p

q = 0#

Else

p = 0#

normp = p

q = 1#

End If

Return

End If

‘***** |Z| <= 37.

expntl = Exp(-0.5 * zabs ^ 2)

pdf = expntl / root2pi

‘***** |Z| < CUTOFF = 10/sqrt(2).

If (zabs < cutoff) Then

p = expntl * ((((((p6 * zabs + p5) * zabs + p4) * zabs + p3) * zabs + p2) * zabs + p1) * zabs + p0) / (((((((q7 * zabs + q6) * zabs + q5) * zabs + q4) * zabs + q3) * zabs + q2) * zabs + q1) * zabs + q0)

normp = p

Else ‘|Z| >= CUTOFF.

‘p = pdf / (zabs + 1# / (zabs + 2# / (zabs + 3# / (zabs + 4# / (zabs + 0.65)))))

p = (zabs + 0.65)

p = (zabs + 4# / p)

p = pdf / (zabs + 1# / (zabs + 2# / (zabs + 3# / p)))

normp = p

End If

If (z < 0#) Then

q = 1# – p

Else

q = p

p = 1# – q

normp = p

End If

Return

End Function