Solved Problems on Benford’s Law

Q1. Calculate the probability of the occurrence of 2 as the leading digit using Benford’s law

Solution:

Here, value of digit d is 2

Using Benford’s law formula we get

P(2) = log10​(1+1/2)

= log_10​(1+0.5)

= log10​(1.5)

= 0.17609

≈ 17.61%

Q2. Suppose we have 5000 records related to stock. How many records we can expect the starting digit to be 8?

Solution:

Using Benford’s Law we need to calculate probability of occurrence of number 8 as leading digit

P(8) = log_10​(1+1/8)

= log₁₀​(1+0.125)

≈ 0.0512

Expected number of records with leading digit 5

= 5000×0.0512

= 256

Expected Number of records with leading digit 5 is 256.

Q3. Calculate the probability of occurrence of 1 or 4 as the leading digit in the stock market.

Solution:

Using Benford’s Law we need to calculate probability of occurrence of number 1 as leading digit,

P(1) = log₁₀​(1+1/1)

= log₁₀​(1+1)

≈ 0.301

Similarly for 4

P(4) = log_10​(1+1/4)

= log_10​(1+0.25)

≈ 0.0969

Therefore combining two probabilities we get

P(1 or 4) = 0.301 + 0.0969

= 0.3979

= 39.79%

Probability of occurrence of 1 or 4 as leading digit is 39.79%

Q4. Calculate the probability of the occurrence of 5 as the leading digit using Benford’s law.

Solution:

Here value of d is 5

P(2) = log_10​(1+1/5)

= log_10​(1+0.2)

= log₁₀​(1.2)

= 0.0792

≈ 7.92%

Benford’s Law is one of the fundamental laws of mathematics that states, “In a large dataset, approximately 30% of numbers start with a 1 while less than 5% start with a 9.”

In this article, we have covered, Benford’s Law definition, formula, examples and others in detail.