Coincidence, fluke, and serendipity are closely related concepts that refer to something unexpected. Coincidence is surprising because there is no obvious apparent cause. Fluke means something accidentally happening with an unintended consequence and serendipity is used for a fortunate result that is obtained while striving for another objective.

Anyway, such events always amaze and astound people. Like you meet a neighbor when on holiday at the other side of the world, or you think of an old friend you haven't seen in many years and he calls in the next hour. But are such experiences really unexpected? Are they exceptional? Is there perhaps a hidden cause? Can we compute the probability of a coincidence? Such are the questions that are discussed in this book. The myth and math of coincidence. Although there is a very light introduction to probability, and some probabilistic analysis of a few cases, the book is perhaps more a collection of some recognizable or legendary examples of coincidences with reflections on what is actually so coincidental about them.

The book opens with ten examples of such stories, like for example you find a book in a second hand book store that you were looking for and it happens to be the copy you had when you were young, or when you take a taxi in two different cities that happens to have the same driver, and it is known that Abraham Lincoln dreamt about being assassinated in the weeks before he was actually killed. However, although it may be the synchronicity of time and place that amazes us, there can be other connections or perhaps unconscious associations that can explain the coincidence and considerably reduce its spooky character.

Time to introduce what probability of some event means and what the odds are of something happening. Some historical notes on the origin of probability are given starting with Cardano's *Liber ludo aleae*, and later contributions by Pascal and Fermat. The Galton board is used to explain the bell shape of the Gaussian distribution. It was Jacob Bernoulli's book *Ars conjectandi* that can be considered the first book on probability and in which the famous law of large numbers was described for the first time. A not-very-mathematical introduction is then given to explain expected value, mean, distribution, standard deviation, etc. Some examples are given about the probability of winning with poker, or the chance that a monkey arbitrarily hitting the keyboard will produce a Shakespearean sentence, or (this is a classic) how large the group should be to make the odds larger than than even that two people in the group have the same birthday. It is also made clear that it is important to realize what the sample space is. The 'enormity of the world' should illustrate how many times the same event happens simultaneously world wide so that the probability of a coincidence becomes much more probable than originally thought. Nonlinear dynamics and the second law of thermodynamics illustrate the seemingly unrelated and unexpected effects a simple modification can produce while the unknown relation between cause and effect could be relatively simple. One should also take some hidden variables into account, i.e., parameters that can couple the two coinciding events but that are not always obvious. If two people incidentally meet at some place and time, this will depend on all the systems these people are part of: social, familial, biological, environmental, political, etc. and all of these can have highly nonlinear chaotic influence on their meeting. Anyway, Mazur applies and simplifies some of this knowledge needed to analyze how probable the 10 cases, that were introduced at the beginning of the book, can actually be. Or rather Mazur makes several assumptions that allow him to justify whether such an event is 'rather probable' and others are 'extremely improbable'.

The previously described contents is making up the first three parts of the book: the ten cases, the mathematics needed, and the analysis of the cases. A rather extensive fourth part is called 'the head-scratchers'. Here probability and analysis are not the main topic anymore. Diverse topics are discussed. It is first explained what a DNA analysis in a criminal investigation really means and it turns out that DNA identification is not as irrefutable as the media want us to believe and innocent people may have been condemned or imprisoned based on disputable evidence. I think Mazur makes a strong point here.

Some scientific discoveries were accidental. Famous are the discovery of penicillin, of X-ray fluorescence, and quinine as an antimalarial, but also Turing and his team decrypting the German Enigma code did have some luck. We learn along the way the origin of the famous dictum, usually attributed to Newton: `If I have seen further it is by standing on ye shoulders of Giants'. It has been used by him, but he was not the first.

Then there is the factor risk that has to be dealt with in many situations. Certainly in gambling but also on the financial markets (which is about the same as gambling), but there are also risks of disasters like earthquakes that are still highly unpredictable.

Under the title 'Psychic power' a discussion is given about ESP. There are obviously many frauds, but Mazur seems not to completely exclude the possibility of some 'action at a distance' because of some electromagnetic or other waves we may perhaps not fully understand yet.

Another chapter discusses some coincidents in the literature. Two examples are discussed in detail: There is the Middle English poem of *Sir Gawain and the green knight*, one of the best known stories of the Arthurian legends, and the story of *The three princes of Serendipity*. The latter Indian story also dates from a 14th century and is brought to Italy in the 16th century via the Persians. It is actually the origin of the word serendipity in English. Fictional coincidences may be the result of subconscious associations made by the author while writing.

To summarize, the mathematics of this book are low level, so the 'math' in the title should not frighten any potential reader. On the other hand there is not a strict systematic analysis that can tell you that such and such surprising event could happen with such exact probability. There are too many factors and unknowns, perhaps hidden relations, that make an event happen. There are of course all the laws of physics that guarantee that in principle every event is completely defined by its past so one could even bring a discussion about free will into the argumentation. Fact is that we do not completely understand some coincidences, or certainly we do not know enough to analyze them with certainty via a mathematical methodology. So, if you are a mathematically interested reader, attracted by the 'math' in the title, the content may be a bit disappointing. Strict analysis is not that simple, except for constructed examples. But isn't that exactly why we find these coincidences so fascinating and that is thus probably why one will want to read this book. Mazur just provides elements that will start you thinking about there phenomena. Some may become less freaky, others we may perhaps never understand completely. Nevertheless it will help you distinguish bogus science like spiritism and the likes of it from what is scientifically possible.

Joseph Mazur is an emeritus professor of mathematics who has written several books on popular mathematics in the last decade. See for example the review of *Enlightening symbols* in this database. He should not be confused with Barry Mazur, another prolific writer of popular math books.