Monday, May 28, 2012

The First-Digit Phenomenon

I just finished a fascinating book by Mario Livio called The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number. In the last chapter “Is God a Mathematician?” one of many surprises Livio offers up makes me want to run over to a local gambling den, or to the local temple (where prayers are believed to be answered by the gods using blocks similar to dice). I would then challenge those present to ask their gods the answer to a question while I consult the “Mathematician” God.

This would be the challenge: choose any set or sets of random numbers, such as a listing from a table in the World Almanac, a chart listing death tolls from major earthquakes, the population of places in given states exceeding 5,000 or more, the numbers listed out on the front page of your daily newspaper in a week, or any combination of anything like the above. What will be the probability that the first digit of any given number is  1,2,3,4,5,6,7,8 or 9?

Common sense tells us that the numbers 1-9 should occur with the same frequency among the first digits. Right? Wrong!

Benford’s law states that the probability P that digit D appears in the first place is given by the equation P = log (1 + 1/D).

That means the probability of a 1 would be about 30 percent; 2, about 17.6 percent; 3, about 12.5 percent, all the way down to 9, about 4.6 percent. Counter-intuitive, isn’t it?

Some lists of numbers do not obey this law (for example, numbers in telephone books where the same few digits repeat in any given region).