Question

It is a striking fact that the first digits of numbers in legitimate records often follow a distribution known as Benford's Law, shown below.

First digit	1	2	3	4	5	6	7	8	9
Proportion	0.285	0.186	0.126	0.084	0.069	0.067	0.034	0.039	0.110

Fake records usually have fewer first digits 1, 2, and 3. What is the approximate probability, if Benford's Law holds, that among 1153 randomly chosen invoices there are no more than 670 in amounts with first digit 1, 2, or 3? (Round your answer to four decimal places.)

Answer 1

P(X = 1, 2 or 3) = 0.285 + 0.186 + 0.126 = 0.597

The number of invoices with first digits 1, 2, or 3 out of the total 1153 randomly chosen invoices is modelled here as:

This is approximated to a normal distribution as:

The probability here is computed as:

Applying the continuity correction, we get:

P( X < 670.5 )

Converting it to a standard normal variable, we get:

Getting it from the standard normal tables, we get:

Therefore 0.1420 is the required probability here.