Fluke: The Math and Myth of Coincidence by Joseph Mazur

Source of book: Borrowed from the library

I first borrowed my dad’s copy of The Hitchhiker’s Guide to the Galaxy when I was, well, pretty young. I didn’t ask permission, and was a slight bit shocked at some of the language (which means I must have been young indeed…) but I was very much hooked.

I mention this because, as all fans will know, a key plot point in the book is the “Infinite Improbability Drive” at the heart of the spaceship, The Heart of Gold. As Douglas Adams explains, “finite improbability” drives had been invented long before, so that tricks like removing a woman’s clothing by random chance had been done for centuries. But it was the ability to do things that were not merely improbable - and with known probabilities - but things for which one could not even calculate the many variables, that made the ship so useful. 

Joseph Mazur, a math professor and author, attempts to put the realm of probability into perspective for the layperson. His attempt is partially successful, which is impressive given the vast scope of such an undertaking, and the difficulty in making it useful for those of us without the ability to do higher math and upper level research. (And by that, I mean beyond algebra and statistics - about my comfort level.) To do this, he limits his topic significantly, and focuses on a few specific scenarios that are likely to be relevant to real life and to the stories of remarkable coincidence that make up our folklore.

To start with, Mazur quotes the dictionary definition of “coincidence,” which most of us probably have not actually thought about.

“A surprising concurrence of events or circumstances appropriate to one another or having significance in relation to one another but between which there is no apparent causal connection.”

As the author points out, several of the elements of a true coincidence are often missed when discussing seemingly chance occurrences.

First, it is important to note that these are events which have some significance in relation to each other. In other words, we don’t notice these occurrences unless they have some significance or connection. In practical terms, then, “coincidences” in the colloquial sense happen all the time, all around us, but they are not noticed because they do not trigger a response in us.

As a practical example of this, chances are you and I have both seen the same person in different circumstances. Had that happened with a friend or someone who stood out, we would call it a remarkable coincidence. Instead, because the person had no personal meaning to us, we never noticed.

The second is the phrase “apparent causal connection.” Let me pause for a moment to appreciate the use of “causal,” one of my favorite words from law school. The point the author makes with this is that many things which seem to be mere coincidences actually have causes that make them much more likely.

For instance, if I were to tell you that I would have a case in front of the same judge that presided over Michael Jackson’s estate, you might (if you knew nothing about me) find it to be a striking coincidence. After all, what are the odds that two people would have a case in the same courtroom at about the same time?

Well, for two specific random people, perhaps it would be rare (although, as Mazur points out, on a population basis, it would be inevitable.) But I am not a random person. I am a California attorney, and I take cases that are heard in the Probate division. Furthermore, while I don’t do much in Los Angeles County, I live just one County over, so I do have cases there. So the chances that a person like me might have this “coincidence” are actually higher than one might think. Because there are many causes for the “coincidence”: I am an attorney in the area who practices Probate.

This also leads to another point that Mazur makes. While the odds of a particular occurrence happening to any one individual (like you…), over large populations, the odds of a particular thing happening increase greatly.

The example that Mazur uses is of a woman who won the lottery four times in 18 years. What are the odds of that? Well, the odds of her winning the lottery that many times in that time frame are indeed astronomically low. So don’t bet your life’s savings expecting to do the same.

However, in 18 years, given the total number of lotteries and players, it is actually highly likely that someone would win this many times. It is even more likely, too, given human behavior - and economics. She almost certainly played more once she won the first time. She had motive, and money to spend…

The best part about this section of the book is the math. Okay, at least in my opinion. Mazur makes the basics fairly easy to understand, at least if you can do algebra. (I love algebra, for what that is worth, even though I never took calculus.) Once you have the basics down, you can figure out probabilities. Then, when you start adding those probabilities together, the likelihood of “rare” occurrences starts to approach a value of one.

This is essentially what powers The Heart of Gold. If you can take improbable events, and stretch the time and number event horizons to infinity, the likelihood (in theory) becomes one, and the event occurs.

Let me also note with approval his use of both coin flips and dice rolls to illustrate the concepts. My brother invented a version of “fantasy” baseball (more or less) based on dice probability - back when he was in grade school. While it could have used some refining, he wasn’t too far off in correlating the odds of specific batting outcomes with the odds of rolling certain numbers with a pair of dice.

The book starts with ten stories of “coincidence” that are then analyzed. Some turn out to be indeed unusual, while others turn out to be highly probable, for the reasons above, and only have significance because the persons involved noticed - and told their stories.

There are some other interesting chapters in the second half of the book, which I thought I should mention.

First is one on DNA evidence in criminal trials. As a lawyer, this has been a fascinating topic, not least of which is because I entered law school not long after the O. J. Simpson trial. (As a former Los Angeles resident at the time, I also knew a good bit about the persons and places involved.)

Mazur points out what most of us attorneys know: DNA evidence is not a panacea, and its validity depends very strongly on the way it was processed and handled. Even if this is perfect, DNA evidence is either highly useful, or not particularly useful at all, and the specifics of when and why are poorly understood by legal professionals and law enforcement, to say nothing of the general population.

Just to give some highlights - which is what Mazur does - from a larger field. (I’ve done a bit of research on my own.) First, there is a non-trivial chance of a false positive or a false negative for DNA testing. Mazur points out, however, that the implications of one are different than the other. The chapter opens with Maimonides’ quote that it is better to let some guilty people go free than to have a single innocent executed. (That this is lost on many in our own age is all too true.)

But it gets worse. Mazur cites an independent investigation of the Houston PD crime lab, which found that 32% had major mishandling issues - and the investigators believed that intentional scientific fraud was the cause. (In other words, police workers manipulated results to get convictions.)

And then it gets even worse, at least from the scientific point of view. Fraud is horrible. But fraud can be fixed. Not so much for systemic problems. Let me see if I can summarize this in an understandable manner. Let’s say that we do DNA testing on a person who is already a suspect for legitimate reasons. (Like, say, motive, opportunity, lack of alibi, and so on.) The chances of a false positive for that person are pretty low. DNA might well be useful evidence here. (Although chance contamination is also a possibility, so it isn’t conclusive, just another piece in the puzzle.)

But what if, instead, there are no suspects, but law enforcement runs all their possible DNA samples to look for a match? Well, now we have a problem. Once you invoke the laws of large numbers, the chances of a false positive increase really fast. With enough “suspects,” it becomes almost certain that there will be a match - a false match.

As Mazur points out, this is particularly a problem in the United States. Believe it or not, while we have a mere 4.4 percent of the world population, we have nearly ¼ of the total prisoners. Let that sink in. Our incarceration rate is the highest in the civilized world. Higher than such hotbeds of “freedom” as Russia and Rwanda. And each of those prisoners has given a DNA sample to be checked. And then think about two more things: we disproportionately incarcerate African Americans; and the chances of a false positive increase with consanguinity or membership in an ethnic group. In other words, the system is set up to frame certain groups of people, and the laws of probability demonstrate the risk. The average jury will not hear this, as you might well imagine.

Just for contrast, DNA evidence does appear to be relatively reliable for one thing: exonerating suspects. The fact that there has been so much resistance from law enforcement to organizations who wish to re-evaluate old convictions is a bit disturbing to me as well.

This review has already run long, so let me mention just a few other things. First, I do think a weakness of the book is that it lacks focus. I think the author would have loved to have gone into depth in a few more areas, covered the math in detail, and made this a far longer book. Instead, he had to pick and choose, and the choices show. There are several of the stories he starts with that I would have loved to have examined in more detail, but space permitted only a cursory look at the ideas, rather than the actual math. Most readers might disagree, however, having glazed over after the one chapter dedicated to the math itself.

This is related to a vague dissatisfaction that I myself felt - and which has occurred to other reviewers. While this book can give some clarity about some aspects of chance, so much still boils down to a combination of hard math and the immense difficulty of teasing out the variables which affect chance. One is left with the feeling that, while things like lotteries are mathematically clear, much of what we experience in life defies easy analysis. The variables are just too many, and the math quickly becomes impossible.

The point of the math, though, is that some things may seem unlikely, but in fact are extremely likely. Others may remain more mysterious. And telling the difference is indeed important. What I might draw from this - and from the fact that variables are hard to determine - is that one needs to be careful not to ascribe to fate (or whatever supernatural being you prefer) what may well be chance. This doesn’t mean one cannot see the hand of God in things, but that one needs to be very careful. Yesterday’s coincidence may well turn out to be tomorrow’s foreseeable result.

Mazur clearly loves his topic, and has thought through the problems presented by coincidence more than most of us ever will.

The questions still remain for us to ponder: when we see something as a truly remarkable coincidence, is it truly mathematically rare? Or it is not so much the workings of an inscrutable fate as the inevitable result of known causes working with the laws of mathematics? For that matter, does it matter to us personally? And how should it shape our public policy? (Yes, the lawyer can’t leave that one out…)

It’s an interesting book. I recommend it, but urge readers to sit down and learn the math. It is the most essential part of the issue.