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 .

Fraudulent numbers in tax returns, payment records, invoices,
etc. often display patterns that aren’t present in legitimate
records. It is a striking fact that the first digits of numbers in
legitimate records often have probabilities that follow the model
(known as Benford’s Law) partially shown in the following
probability distribution, where the random variable x is the first
digit of the number. x 1 2 3 4 5 6 7 8 9 P(x) 0.301 0.176 0.125 ?
0.079 0.067 0.058...

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...

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
0.28
0.152
0.129
0.088
0.05
0.07
0.042
0.035
0.154
Fake records usually have fewer first digits 1, 2, and 3. What
is the approximate probability, if Benford's Law holds, that among
1189 randomly chosen invoices there are no more than 688 in amounts
with...

Recall that Benford's Law claims that numbers chosen from very
large data files tend to have "1" as the first nonzero digit
disproportionately often. In fact, research has shown that if you
randomly draw a number from a very large data file, the probability
of getting a number with "1" as the leading digit is about 0.301.
Now suppose you are an auditor for a very large corporation. The
revenue report involves millions of numbers in a large computer
file....

Recall that Benford's Law claims that numbers chosen from very
large data files tend to have "1" as the first nonzero digit
disproportionately often. In fact, research has shown that if you
randomly draw a number from a very large data file, the probability
of getting a number with "1" as the leading digit is about 0.301.
Now suppose you are an auditor for a very large corporation. The
revenue report involves millions of numbers in a large computer
file....

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