What is the probability that the leading digit is 3? What is the probability tha

Question

What is the probability that the leading digit is 3?

What is the probability that the leading digit is at least 7?

While reading through this year's Arkansas Democrat-Gazette newspapers, you record all numbers appearing in a news article-the number of wins by a football team, the total charitable contribution of a politician, the income of a CEO, and so on. Now consider the leading digit of each number. Intially, you might think the leading digit of a randomly selected number would be equally likely to be any of the values (a discrete uniform distribution). However, evidence suggests an alternative probability distribution called Benford's Law: fx(x) = P(1st digit is x)= log 10(1 +-), x = 1,2, ,9

Explanation / Answer

What is the probability that the leading digit is 3?

We have to find P(X=3)

P(X=x) = log10(1 + (1/x)), where x = 1 to 9

P(X=3) = log10(1 + (1/3))

P(X=3) = log10(1 + 0.3333)

P(X=3) = log10(1.3333)

P(X=3) = 0.124928

Required probability = 0.124928

What is the probability that the leading digit is at least 7?

We have to find P(X7)

P(X7) = P(X=7) + P(X=8) + P(X=9)

P(X=7) = log10(1 + (1/7))

P(X=7) = log10(1.142857)

P(X=7) = 0.057992

P(X=8) = log10(1 + (1/8))

P(X=8) = log10(1.125)

P(X=8) = 0.051153

P(X=9) = log10(1 + (1/9))

P(X=9) = log10(1.111111)

P(X=9) = 0.045757

P(X7) = P(X=7) + P(X=8) + P(X=9)

P(X7) = 0.057992 + 0.051153 + 0.045757

P(X7) =0.154902

Required probability = 0.154902

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