### Introduction

At least once or twice, we experience uncanny coincidences. The friend who called us before we picked the phone to rang them or we bump our neighbor overseas. This seemed unlikely to happen but when experienced, life seems less random and the world seems smaller. A popular story of coincidence is told by the novelist Anne Parish.

At Paris in 1929, Parish was strolling at a book stall in Seine. An old favorite,

Jack Frost and Other storiesgrabbed her attention. When she turned the cover, she saw "Anne Parrish, 209 North Weber Street, Colorado Springs, Colorado” which is inscribed on the opening page. It turned out that the book was her childhood copy.

I came across Parrish's story in Joseph Mazur's Fluke. Stories of coincidence such as Parrish is an interesting story to tell and brings out some questions such as "Are these event random or predestined?". These stories sparked researchers to dig on coincidence and one of them is Joseph Mazur, a mathematician. In Fluke, Mazur used probability to uncover the chance of an events (coincidence) to happen. He defines the boundary between a coincidence and a fluke. Coincidences are as meaningful event that happened without any cause while flukes are improbable events with a clear cause. For example, winning a lottery is not a coincidence but a fluke hence we need to bet first to win. A case of recurring coincidence is foretold by the tale of Emile Deschamps and the plum pudding.

Deschamps was given a plum pudding (an unusual dish in France) by a Monsieur de Fortgibu at a boarding school in Orleans. After a decade, Deschamps wandered down in a street of Paris and saw a plum pudding on the menu of a restaurant. He decided to dine and order the plum pudding. The last slice was taken by Monsieur de Fortgibu who is willing to share it. Years later, Deschamps is dining in a friend's home when a plum pudding is announced to be served. He wonders whether he would meet Monsieur de Fortgubu. The door bell rang and Monsieur de Fortgibu arrived. It turned out the hostess is not expecting him. Monsieur de Fortgibu supposed to dine at another house that night but rang the wrong doorbell.

Deschamps' encounter with Monsieur de Fortgibu after years and on different places is not the same coincidence as to the Parish's story. Mazur stated that coincidences are not created equal. Some of these stories are a byproduct of shared travel, class and communication. In Fluke, there are ten stories of coincidence and Mazur elaborate the chance and odds for coincidence to happen. Mazur makes several assumption to justify the event as *rather probable* or others as *highly improbable*.

In Method of Studying Coincedence, the mathematicians Persi Diaconis and Frederick Mosteller defined: coincidence is a surprising concurrence of events that are meaningfully related but without apparent causal connection. Statistician David Hand in The Improbability Principles, agued that in pure statistics, these events are random and without meaningful relations. He added coincidence shouldn’t be that surprising since this events happen all the time. His argument is quite valid hence we are liberated in considering which is coincidence. Meeting a person who shares a birthday seems like a fun coincidence. Also, we have the same feeling as to a person who shares our mother's birthday or a day right before or after ours. We feel coincidental to several birthdays.

### The Law of Large Numbers

With over 7 billion people, any outrageous thing is likely to happen. Mazur, in fact, stated that we need to understand the law of large numbers. **Weak law of large** numbers explained that the average outcome is closer to the expected value as more experiments are performed. For example, The probability of getting a head or a tail in flipping a coin reaches close to 0.5 after a coin is flipped a million times. The weak law has a better grasp on handling uncertainty.

In contrast, the **Law of Truly Large Numbers**, elaborates that it is probable for any outrageous event to happen with large enough sample. If enough people buy tickets, there will be a lottery winner. To the person who wins, it’s surprising and miraculous, but a person winning a lottery doesn’t surprise the rest of us. The opportunities of coincidence to happen is probable even with limited sample of our lives. For example, when we consider the people and places we knew and there is a good chance of meeting a person we know.

Deschamps' story can be explained by the law of truly large numbers but lacks quality. Deschamps' coincidence boils to a meagre: *"it was possible so it happened"*. It does a good job in explaining Joan Ginther's case.

Joan Ginther is Texas woman who won four multimillion-pound lottery jackpots over 18 years.

Ginther's winning seems unbelievable but considering a huge sample, the winning is likely to happen. The US has 26 lotteries and 104 draws per year. If we assumed about 320 million Americans bet on lottery every week, Mazur works out that Gither's odds are better than a person to win twice in five year. With vast sample size, the probability of winning four times is close to 1. Luckily, it is Ginther.

### The Probability of Coincidence

Mathematicians liked to demonstrate how common unlikely-seeming events can be. They used the birthday problem.

The question is how many people need to be in a room before there’s a 50/50 chance that two of them will share the same birthday.

We find the the birthday problem as an impressive tool in thinking about coincidences. We suppose N as the number of people in the room and c are the birthdate where N is less than or equal to c. The chance of no match in any categories is given by

The odds of not having a match is e^{(-N2/2c)}. The probability of having at least one match is defined as *(1-p)* that returns N = 1.5 divided by square root of c or 2.5 divided by square root of c. If we consider c = 365, we have N = 22.9 or 23 for a 50% chance and about 48 for a 95% chance.

The probability of coincidence is solved by looking into the rates of occurrence (frequency of coincidence) of two independent events. For example, we evaluate the odds of a friend called us at after we thought of them. We count the number of times we thought about them and divided it by the total number of thoughts. Assuming that over the years, the rate of this though is about 0.005 or 5 out of 1000 thoughts. Similarly, we estimate the contacts from that friend about 0.01 or 1 out of 100 contacts. Then, the probability of coincidence is 0.05 multiplied to 0.01 which is equal to 0.0005. The probability is quite low but probable with larger samples.

Bernard Beitman coincides with Mazur in affirming that coincidence are not created equal. He stated that the approach to probability of coincidence is different to each coincidental event. He added that a smaller time window makes coincidences less probable and reason for contact other than saying a "Hi" makes it less probable.

Calculating the probability of coincidence is a daunting task hence there are too many constraints to consider. If we include the time window and personal meeting, the probability model becomes complex. Also, measuring the base rates of coincidence are quite difficult hence it is intangible.

### Final Thoughts

Parrish's story and the other stories of coincidence are delight to hear and tell. Many have tried to explained coincidence with math and probability but we are still far from understanding coincidence. Hence coincidences are not created equal, there are more constraints to fully account. The law of large numbers may explained the occurrence of some coincidental events but not all. Above all of these, coincidences are still an open question.

### References

- David Shariatmadari, Fluke: The Maths and Myths of Coincidences by Joseph Mazur – review
- Alexander Woollcott, Reunion in Paris
- Julie Beck, Coincidences and the Meaning of Life
- Persi Diaconis and Frederick Mosteller, Method of Studying Coincidence
- David Hand, The Improbable Principles
- Coincidences Have More To Do With Math Than Magic
- What A Mathematical Formula Can Teach Us About Coincidence
- Bernard D. Beitman M.D., The Promise & Problems of Probability in Coincidence Studies
- Coincidences: What are the chances of them happening?