‘What are you doing here?”

“Everybody’s got to be somewhere!”

This exchange (as I recall, from the BBC Radio “Goon Show” in the 1950s) illustrates the first of the fundamental concepts espoused by David J. Hand in his latest book, “The Improbability Principle.” The rest of the title explains it thus: “Why Coincidences, Miracles, and Rare Events Happen Every Day.”

According to the author of this very engaging book, there are five strands (or “Laws”) contributing to the Improbability Principle: the Law of Inevitability, the Law of Truly Large Numbers, the Law of Selection, the Law of the Probability Lever and the Law of Near Enough. So what is the Improbability Principle?

Hand succinctly states it as follows: Extremely improbable events are commonplace. Indeed, in 1866, the British mathematician Augustus De Morgan wrote, “Whatever can happen will happen if we make trials enough.” Hand summarizes this as: Someone, somewhere, at some time won a lottery twice, and he provides several examples of this, along with an account of a man who survived seven different strikes by lightning. Perhaps, though, this is not quite as amazing as it first appears; Roy Sullivan, a park ranger in Virginia, no doubt spent a great deal of time outside in all kinds of weather. He was struck in 1942, 1969, 1970, 1972, 1973, 1976 and 1977, and also, he claimed, once as a child.

’The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day" by David J. Hand (Macmillan )

No doubt most people have experienced what we like to call amazing coincidences. The first chapter chronicles one such event, in fact, a series of events involving the actor Anthony Hopkins and the George Feifer novel “The Girl From Petrovka.” I will not spoil the impact of the story by describing it here. Something less amazing, but nevertheless an odd coincidence, occurred when I started reading Chapter 2, “A Capricious Universe.” Hand quotes a verse from the book of Proverbs: “The lot is cast into the lap, but its every decision is from the Lord” — a passage I had read the evening before!

Basically, probabilities and chance are counterintuitive. We tend to underestimate high probabilities and overestimate low ones. The birthday problem is a delightful example of this. What is the lowest number of people who must be in the same room to make it likelier than not that at least two of them have the same birthday (day and month, not year)? The answer is a surprisingly small 23. I have used this to great effect in classrooms as an icebreaker. If there are 30 students, the probability of a shared birthday is about 0.7 (or 70 percent). Since my birthday is in December, the interest builds as we proceed through the year month by month, and on several occasions I have been one with a shared birthday.

One common source of error in our understanding of events around us is the confusion of correlation with causation. When my elder daughter was a toddler, she wore a bib at mealtimes to prevent her clothes from getting (too) messed up while she ate. When she was hungry, she would attempt to put on the bib, hoping that food would appear. It was not long before she recognized this error. Placing the bib around her neck did not guarantee the arrival of food any more than swaying trees cause the wind to blow.

There are also other underlying principles at work. We tend to ignore evidence that does not support our theories (and indeed, our political views), emphasizing only that which does. We can also cherry-pick our data. Painting bull’s-eyes around arrows you shot into a door would qualify as selection bias.

Sometimes a little careful thought can, in retrospect, calm feelings of panic induced by some discouraging event or statement. Suppose, for example, a test for some disease is 98 percent accurate, meaning that if someone has the disease, 98 percent of the time the test is positive, and if someone does not, 98 percent of the time the test is negative. If 1 percent of people actually have the disease, and you test positive for it, how worried should you be? If 10,000 people are tested, then about 1 percent of those (or 100 people) have the disease, and 9,900 do not. Of that group of 100, 98 percent (or 98) will correctly test positive. Of the disease-free 9,900, 2 percent (or 198) will falsely test positive. So there are 296 (98 + 198) positive results, of which 198, or about two-thirds, are false. So you should be cautiously optimistic about not having the disease (but get tested again anyway).

As Hand writes, “The Improbability Principle tells us that events which we regard as highly improbable occur because we got things wrong. If we can find out where we went wrong, then the improbable will become probable.” I am reminded of the story of two gentlemen in a Dublin pub drinking to the amazing coincidences that keep piling up as they recount their early lives to each other. When another patron enters and asks what’s going on, the bartender merely points out that the O’Reilly twins are getting drunk again.

In summary, if you wish to read about how probability theory can help us understand the apparent hot hand in a basketball game, superstitions in gambling and sports, prophecies, parapsychology and the paranormal, holes in one, multiple lottery winners, and much more, this is a book you will enjoy. I will go further. The statistician Samuel S. Wilks (paraphrasing H.G. Wells) said that “statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.” With that laudable goal in mind, “The Improbability Principle” should be, in all probability, required reading for us all.

John A. Adam is a professor of mathematics at Old Dominion University and the author of several books, including “X and the City: Modeling Aspects of Urban Life.”

THE IMPROBABILITY PRINCIPLE

Why Coincidences, Miracles, and Rare Events Happen Every Day

By David J. Hand

Scientific American/Farrar Straus Giroux. 269 pp. \$27