Bascule is reading a book. What is the probability that the first digit of the page he is on is 1, 2, 3, 4 or 5?

a) obtain an expression
b) approximate a numerical value

This article is relevant: http://en.wikipedia.org/wiki/Benford%27s_law. Although referred to as a law, it is typically only weakly supported and purports to cover quite different phenomena - In the book page numbering problem (similar situations occur with street numbering, etc) we'd expect lower digits to occur more frequently, since the likelihood with which a given integer makes an appearance decreases with the size of the number. (Lower numbres appearing in all sets, higher numbers only in larger sets, and then less often). Presumably to "prove" Benford's law for such cases one would like to have a reliable distribution of the size of books..

A less transparent case of Benford's Law concerns, e.g. the frequency of digits bank account balances or the price of common consumer items in whatever currency. In the latter case, we could suppose that the value of a unit of currency would naturally be pegged (at least initially) for convenience to typical prices of common consumber items (approximately 1 currency unit for an apple or a bread roll). Whereas inflation will distort this relationship over time, excessive inflation gives rise to public demands for recalibrating the currency. (This part of the discussion hardly bears on the problem, but is included for readers who might find it interesting). 

 

Posted by FrankM on 2008-01-06 22:30:40