EXAMPLE 12 Benford’s Law: One Is the Likeliest Number You’ll Ever Know

Faked numbers in tax returns, invoices, or expense account claims often display patterns that aren’t present in legitimate records. Some patterns, like too many round numbers, are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the first (leftmost) digits of numbers in legitimate records often follow a model known as Benford’s law, which is shown in Table 8.6. (Note that a first digit can’t be 0).

You should check that the probabilities of the outcomes sum exactly to 1 to verify that this is a legitimate discrete probability model. Using this model, investigators can detect fraud by comparing the first digits in records such as invoices paid by a business with these probabilities. For example, consider the events = “first digit is 1” and = “first digit is 2.” Applying Rule 4, the addition rule for disjoint events, to the table of probabilities yields , which is 0.477 (almost 50%). Crooks trying to “make up” the numbers probably would not make up numbers starting with 1 or 2 this often.