Dr. Diaconis and Dr. Mosteller said they decided to study coincidences because they were fascinated by the role these odd events play in everyone's lives. ''All of us feel that our lives are driven by coincidences,'' Dr. Diaconis said. ''Who we live with and where we work, why we do the things we do often rest on slim coincidences.'' These chance events ''touch us very deeply,'' he said. The two statisticians defined a coincidence as ''a surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection.''

Dr. Diaconis and Dr. Mosteller began with the presumption that there are no extraordinary forces outside the realm of science that are acting to produce coincidences. But they also recognized that seeming coincidences are an important source of insight in science and so should not be dismissed out of hand. What looks like a coincidence may in fact have a hidden cause, which can lead to a new understanding of a phenomenon. A sequence of odd blips on a chart, a clustering of cases of a rare disease can tell researchers that a new event is occurring.

Anthologies of Happenstance

A decade ago, Dr. Diaconis and Dr. Mosteller started asking their colleagues, friends and friends of friends to send them examples of surprising coincidences. The collection quickly mushroomed. Dr. Mosteller said he had 13 notebooks, each three and a half inches thick, full of coincidences. ''These notebooks are eating up the shelves in my den,'' he said. Dr. Diaconis said he had 200 file folders full of coincidences.

When they began to study these coincidences, they learned that they fell into a several distinct groups. Some coincidences have hidden causes and are thus not really coincidences at all. Others arise from psychological factors, like selective memory or sensitivities, that make people think particular events are unusual whether they are or not. But many coincidences are simply chance events that turn out to be far more likely statistically than most people imagine. The analyses often required the researchers to develop new statistical methods, but in the end almost all coincidences could be analyzed.

The law of truly large numbers, which explains the double winner of the New Jersey Lottery, says that even if there is only a one-in-a-million chance that something will happen, it will happen eventually given enough time or enough people. ''It's the blade-of-grass paradox,'' Dr. Diaconis said. ''Suppose I'm standing in a large field and I put my finger on a blade of grass. The chance that I would choose that particular blade may be one in a million. But it is certain that I will choose a blade.'' So if something happens to only one in a million people per day and the population of the United States is 250 million, ''you expect 250 amazing coincidences every day.''

''If a one-in-a-million thing happens to you, you start telling people about it,'' Dr. Diaconis went on. ''You might say to me, 'So what do you think of that, wise guy?' And I say, 'It's an example of the law of truly large numbers.' ''

Right Answer, Wrong Question

When a New Jersey woman won the lottery twice in a four-month period, it was reported as a one-in-17-trillion long shot. Narrowly speaking, that is correct. But as Dr. Diaconis and Dr. Mosteller reported, one in 17 trillion is the odds that a given person who buys a single ticket for exactly two New Jersey lotteries will win both times. The true question, they say, is, ''What is the chance that some person, out of all the millions and millions of people who buy lottery tickets in the United States, hits a lottery twice in a lifetime?''