The Number Sense - Benford's Law

"Would you like to try a bet?

Open a book (any book) at random and note the first digit that your encounter.

If this digit is either 4, 5, 6, 7, 8, or 9, you win $10.

If this difit is either 1, 2, or 3, I win $10. "

Do you think your winning odds (chances) are higher than mine?

I read this in the booked " The Number Sense: How the Mind Creates Mathematics" by Stanislas Dehaene. (Published in 1999, p. 113)

If we keep on playing for a long period, say the running time of a typical movie (2 hours), you would almost always lose. We shall look at the statistics researched by many scientists and mathematicians.

Benford's law (named after physicist Frank Benford), also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way.

Measurements of real word values are often distributed logarithmically (or equivalently, the logarithm of the measurements is distributed uniformly).

This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). The result holds regardless of the base in which the numbers are expressed, or the units the measurements are taken, although the exact proportions change.

Income tax controllers use them to check those who are dishonest about their declaration. Benford's Law is also employed to check possible fraud cases in general election, e.g. 2009 Iranian Election.