Hand Calculations-Formulas by H.Wieman/Gene VB/Victor P./
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I have attached pdf files of these pages at the bottom.
The point to begin about the derivation of 'Probability of finding the correct hit' is here: www-rnc.lbl.gov/~wieman/HitFinding2DXsq.htm
Once you are there you can follow (backwards) the whole history through the embedded links.