At the meeting of the Mathematical Association of America held in the Summer of 1955, I had the privilege of delivering the Hedrick Lectures. I was highly gratified when, sometime later, Professor T. Rado, on behalf of the Committee on Carus Monographs, kindly invited me to expand my lectures into a monograph.

At about the same time I was honored by an invitation from Haverford College to deliver a series of lectures under the Philips Visitors Program. This invitation gave me an opportunity to try out the projected monograph on a “live” audience, and this book is a slightly revised version of my lectures delivered at Haverford College during the Spring Term of 1958.

My principal aim in the original Hedrick Lectures, as well as in this enlarged version, was to show that (a) extremely simple observations are often the starting point of rich and fruitful theories and (b) many seemingly unrelated developments are in reality variations on the same simple theme.

Except for the last chapter where I deal with a spectacular application of the ergodic theorem to continued fractions, the book is concerned with the notion of statistical independence.

This notion originated in probability theory and for a long time was handled with vagueness which bred suspicion as to its being a bona fide mathematical notion.

We now know how to define statistical independence in most general and abstract terms.