Question

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 digits of numbers in legitimate records often follow a model known as Benford’s law. Call the first digit of a randomly chosen record X for short. Benford’s law gives this probability model for X (note that a first digit can’t be 0): Let A = {first digit is at most 3} and B = {first digit is odd}

a) What is the probability of A?

b) What is the probability of B?

c) What is the probability of A or B?

d) Why does the probability of A or B not equal the sum of the probability of A and the probability of B?

Answer 1

A is  the event that First digit is at most 3 , i.e 1 or 2 or 3

So,

a) P(A) = 3/9 = 1/3 [ As there are 9 possible numbers to chose from , namely 1 to 9 ]

B is the event that  first digit is odd

b) There are 5 odd numbers in the range 1-9

So, P[B] = 5/9

c) A or B : Either the first digit is at most 3 or it is an odd number

ie. the number is 1,2,3,5,7,9

So, P[ A or B] = 6/9 = 2/3

d)   because A and B are not mutually exclusive. The number 1 and 3 are common in both the events .