Coincidence

People are entirely too disbelieving of coincidence. They are far too ready to dismiss it and to build arcane structures of extremely rickety substance in order to avoid it. I, on the other hand, see coincidence everywhere as an inevitable consequence of the laws of probability, according to which having no unusual coincidence is far more unusual than any coincidence could possibly be. A coincidence is a series of two or more chance events that appear to be causally related but really aren't. Many forms of superstition and pseudoscience have their origins in the misinterpretation of said events. The reason for this is a failure to apply statistical thinking to real-life situations, a phenomenon known as magical thinking.

Birthday paradox
According to the birthday paradox, in a room with 23 people there is about a 50% chance that two of them will share a birthday. Most people would think of that as very strange, since 23 isn't even 10%, much less 50%, of 365, the number of days in a year. However, this thinking reflects the self-centered nature of human thought and the usual failure to take the whole statistical picture into account. The birthday paradox states not that there is a 50% chance that any person will share a birthday with someone, but that there is a 50% chance that there is one pair of people in the room sharing a birthday, even though the odds of any specific individual being part of that pair is low. Even more counter-intuitively, in a room with 57 people, there is a 99% chance that a birthday-sharing pair will be in the room. The relevance of this is that some may be tempted to think that if they meet someone who shares their birthday at a party, they must have been brought together by destiny or fate, or some other supernatural force, when in reality, it's just a result of the normal statistical processes of the world.

Paranormal
A person dreams about an acquaintance or friend of theirs whom they have not seen for several years, and about whom they have not thought about for quite some time. The next day, they meet the acquaintance on the street or receive a phone call or email from him. To the untrained mind, this would appear to be so unlikely that it must be evidence of a sixth sense. However, improbability does not mean impossibility. Yes, it's a very unlikely event, but people get many tries for this sort of thing to happen. A person dreams thousands of times over the course of their lifetime, and they are struck by the one coincidence, but forget the vast majority of mundane dreams that had no relation to the events of the day. If you flip a coin thousands of times, you will, eventually, get some very improbable sequence of results, such as 10 heads or tails in a row, merely by chance (this is more or less the reasoning behind Littlewood's law of miracles). If there is only a 1% chance of a dream being a coincidence, then one can expect such a dream to occur once per 100 dreams. If there is a 0.1% chance, then one can expect it will occur around once every 1000 dreams (and most people probably dream at least a thousand times in their lifetime.)

Suppose someone dreams of an earthquake, and the next day, one happens. Now, they might take this to be so unlikely, that, rather than it being a chance event, it must be due to some psychic or intuitional power. They might say, "I've never dreamed about earthquakes before," or only once or twice before, "so what are the odds that it's a coincidence that I happened to dream of earthquakes for the first time exactly when an earthquake was happening?"

But consider the fact that there are 7 billion people on the Earth (even just 1 billion is a mind-bogglingly huge number; just think about it). Out of this 7 billion, perhaps 1 billion might dream per day. It seems a not unreasonable assumption that out of these 1 billion people, every single day some number (perhaps hundreds or thousands) will dream about earthquakes, so that it could very well be said that every single earthquake will be preceded by coincidental "premonitions". Even if the probability of the dream is so low that the number of dreams required is larger than a person will dream in their entire life, that doesn't mean it can't happen by chance. If the odds of a given type of dream happening is one in a million, and one billion people dream every day, then 1,000 people will dream it every day just by chance.

Not to mention that claims of psychic dreams do not have to be specific. A person might say the dream was caused by ESP or "vibrations" or whatnot even if the disaster was an earthquake, and the dream was about some other natural disaster, such as a flood, meteor strike, or volcanic eruption, another event involving many deaths and/or destruction, or even a completely unrelated event that could be interpreted as metaphorically referring to an earthquake. If a certain number of people dream about earthquakes every day, making coincidental dreams certain to happen, then the number of people dreaming about any of these possible calamities is undoubtedly even higher, making the possibility that the dream is a coincidence an even more attractive explanation.

Though from an individual's subjective, bias-filled point of view, a dream might seem so unlikely to be a coincidence that it must be caused by some supernatural presentiment, if one just zooms out and looks not only at all the individual's dreams, but also the broader picture of the enormous number of dreams experienced by all humans daily, it is obvious that such coincidences are in no way unusual, and are, indeed, entirely to be expected. This can be summed up as: Given a large enough number of people or events, unlikely coincidences will happen.

As always, extraordinary claims require extraordinary evidence. All these "proofs" of psychic power can be explained as commonplace coincidences, which, if you look at them statistically, are nothing special, and thus not extraordinary.

Alternative medicine
In the context of alternative medicine testimonials, coincidences occur when a period of natural improvement occurs at the same time as the use of a treatment. This can manifest itself in regression to the mean or spontaneous remission. In a certain type of anecdote, a person had been suffering from a disease for several years (even above 10!) and then got better only a few days or weeks after applying some alternative treatment. How could it be a coincidence? You just have to realize that millions of people use alternative medicine every day, and stories like these are certain to happen because of the extremely large population involved. It must also be remembered that anecdotes like these have been used to justify everything from bloodletting to birthwort. If it were impossible for these coincidences to occur, that would mean both of these treatments (as well as many others) would be effective, which they of course are not.

A person who believes in alternative medicine may attribute any and all improvement to the particular herb or supplement they were taking at the time, and ignore all the times when they took a supplement and did not get better. Of course, frequently, diseases just go away on their own, and the treatment can be credited for the improvement no matter how long it took for the illness to be "cured".

Anecdotal evidence is not acceptable proof for these reasons and more. Statistical studies are required, because they take into account all the events, not merely the ones that strike one as interesting or favorable.

Health scares
Some health scares are caused by the death of a person who shortly before their demise had been vaccinated or had ingested some substance. But some of the substances in question are very widespread. Millions of people get vaccinated, and in some places the vast majority of people are vaccinated. In a given large city, many people die every day from the usual causes, and out of this number, some may have gotten vaccinated on the same day they died of something else. (Even if the death was causally linked, the risks of the intervention must be weighed against the benefits. In other words, the total number of deaths should be compared to the number of lives saved. Many people die of drowning, yet that does not mean the world would be better off without water.)

Conspiracy theories
The frightening nature of coincidences plays a key role in almost any conspiracy theory imaginable. Conspiracy nuts don't like to accept that coincidences happen, so whenever one of their "whistleblower" dies, they immediately assume [INSERT BOOGEYMAN HERE] is to blame, without evidence, of course.

Some have even gone so far as to refer to people who don't believe every Internet crank they see (read: rational people) as "Coincidence Theorists", because "they believe every single suspicious thing that happens is a coincidence.” The problem with this is that there are conspiracy theories that we sheeple believe, such as NSA surveillance programs and Watergate. It's true that governments don't always tell the truth, but not every single attack, crime, natural disaster, and world event is part of a conspiracy.

For example, did some politician happen to cancel a flight scheduled on the same day as a terrorist attack involving airplanes happened? In conspiracy theory land that cannot be a coincidence. There must be a connection.

Or how about the Pentagon having blast-proof windows on one of its walls, and a plane crashing precisely on that wall? Given that the Pentagon has 5 outer walls, the likelihood of this happening is roughly 20%, which isn't actually all that small. One in five isn't very unlikely, except of course in conspiracy fantasy land, where it cannot be a coincidence.

Multiple discovery
(sometimes called "simultaneous invention" or "steam-engine time") is the hypothesis that many if not most scientific discoveries and inventions are made independently and more-or-less simultaneously. The hypothesis contrasts to the which posits that great discoveries are made by geniuses. Important examples of multiple independent discoveries include calculus (Isaac Newton and Gottfried Wilhelm Leibniz), oxygen (multiple scientists), and evolution (Charles Darwin and Alfred Russel Wallace). More examples can be found on Wikipedia: