Poisson Distribution

She would have kited only 18 invoices.

Explanation

The skill of the auditor is impressive, but such knowledge can work both ways. Had Janice known in advance the size of the auditor's sample, she could have limited herself to the largest number that is likelier than not to occur zero times in that sample. The question for Janice would have been: What is that number?

Janice's best chance (consistent with the maximum possible embezzlement) is when the rate p(0), or nondiscovery, is greater than than the rate for p(1) and all higher values put together: the various outcomes that lead to discovery. Since the total amount of probability in a system is always 1, this amounts to saying that p(0) should exceed 0.5. The fastest way to find the smallest such rate is to look for it by scanning along the top row of any Poisson table, which will get us close to the answer, and then calculating values for any necessary nearby points not on the table. To two decimal places, the required rate is r = 0.69, for which p(0) = 0.5106. (By contrast, for r = 0.70, we get the risky value p(0) = 0.4966, and 0 is no longer likelier than all the non-0 outcomes put together. Discovery becomes likelier than nondiscovery).

Knowing the rate, we can now find x, Janice's maximum safe number of kitable invoices. The algebra is:

150x/4000 = 0.69, whence
0.0375x = 0.69, and
x = 18.4

Or shall we say, keeping to the safe side, x = 18. Then by kiting 18 or fewer invoices, Janet will be in a situation where she is more likely than not to evade detection by an auditor using a 150-invoice sample size.

Of course, auditors read books too. Our 150-invoice auditor will very possibly adjust his sample so as to have a better than even chance of detecting peculation at the 18/4000 = 0.0045 rate, and to do so, he too will set about reading along the top row of the Poisson table. It would be funny if he and Janet bumped into each other one day, reading along that row, but from opposite directions.

Comment

It was too much to hope that, with the advantage of our exposition of the problem available to her in early life, Janice would never have turned to a life of crime in the first place. Our mistake was: We shouldn't have mentioned the Goya. Given that tactical error on our part, the most social amelioration that can be hoped for in this case is probably the one sketched in above. How is it likely to end?

In all probability, not in a moral equilibrium, but in an economic one. How much employee theft is a firm better off overlooking? One paper clip? Two? A whole box of paperclips? The law itself recognizes that some amounts are too small to justify court proceedings; the courts have heavy schedules. Again, How much protection, in the form of lengthier audits, can a firm of a given business volume afford?

Somewhere along in here, the problem acquires too many nonmathematical dimensions to be suitable for a textbook. At about the same point, the problem becomes real enough to interest the people who are actually dealing with life. It would seem that a textbook, in its nature, will inevitably fall short of usefulness in real situations. That is largely true. What a textbook can hope to do is to point people in the direction of the real answer, and wish them well on the rest of their journey.