Coincidences and Miracles

COINCIDENCES AND MIRACLES

Coincidences

The most astonishingly incredible coincidence imaginable would be the complete absence of all coincidences.

• John Allen Paulos (from Beyond Numeracy)

We notice certain things and ignore others. We select and talk about the amazing event, not the ordinary ones. We see coincidences after they happen. We don't see them before they happen.

We underestimate how many opportunities there are for "unlikely" events to happen. Surprises and improbable events happen if they have enough opportunities to happen. There are many ways in which events may be linked together.

Getting 5 tails in a row is certain to happen somewhere, sometime to someone. The chances may be small that an event happens at a particular place, time or to a particular person. But with many places, over long periods of time or with many individuals, the seemingly improbable will happen. As Aristotle says: "It is likely that unlikely things should happen."

Someone flips tails 20 times in a row.

A particular run is unlikely, but many independent attempts create more opportunities. If each of 1,048,576 people makes one run, the expected number of successful runs is one, rather than a guarantee of a success. The source uses a historical U.S. population of about 280 million. If each person had one independent daily opportunity for a one-in-a-million event, the expected total would be 280 per day; neither the daily opportunities nor that exact total follows from population alone.

How likely is it that two people share the same birthday?

There are many opportunities for coincidences. For example, in a group of 23 people, the probability is 50.7% that at least two people share the same birthday. It is rather likely that events like this one happen since there are many ways that 2 people can share an unspecified birthday. Observe that the question is not how likely it is that 2 people share any particular birthday. The question is whether it is likely that 2 people share an unspecified birthday.

How many people have to be present at Mary's birthday dinner so there is more than a 50% chance that one of the other guests has the same birthday as Mary? 254 people in total, since we now work with the restriction "two people must share a particular birthday." Both birthday calculations assume independent birthdays with equal probabilities across the days of the year, ignoring leap days. Mary is included in the dinner’s total; the other guests supply the opportunities to match her birthday.

Scale matters. When the numbers are large enough, improbable things happen. The more available opportunities or the longer the time, unlikely events happen. Ask: What are all possible outcomes and their likelihood? What else could have happened?

Making up causes for chance events

Humans are pattern seeking, storytelling animals. We look for and find patterns in our world and in our lives, then weave narratives around those patterns to bring them to life and give them meaning.

• Michael Shermer (Publisher of Skeptic)

Suppose John independently tosses a fair die six times and the result is either (A) 623514 or (B) 666111.

Which alternative showed the correct outcome? Even if (A) looks random and (B) looks like it has a pattern, in independent chance events, both A and B are equally likely or unlikely to appear. Random events may not look random. Independent sequences often show order or streaks. The account cites clusters of bomb hits in London during the Second World War as an example of a pattern compatible with chance. Such an analysis concerns a particular spatial model; it does not mean that all wartime targeting was random.

We want to find reasons for all kind of events - random or not. We search for patterns even where none exist. For example, there must be something important that is happening if a particular number comes up again and again. But it is always possible to find patterns and meaning in an event if we actively search for them and selectively pick anything that fits the pattern and ignore everything that doesn't. But we can't predict the pattern in advance.

John rolls five dice and gets five sixes.

The probability that some roll contains any five specified numbers is very small or one in 7,776 (for example, 6x6x6x6x6). When we toss five dice there are precisely 7,776 ordered outcomes that can appear. With fair, independent dice, every ordered outcome is equally likely and one of them is certain to happen every time we roll five dice. Even if one particular combination (five sixes) is improbable, no single combination is impossible. Any combination that happens is simply one out of a number of equally likely outcomes. It may have been improbable that John rolled five sixes, but not impossible.

More opportunities can make an improbable but possible event more likely to occur. People have seen a human face on Mars, faces in rocks, clouds or even in a grilled cheese sandwich. But that is no mystery. Given the large numbers of rocks, clouds, and sandwiches, sooner or later we will find one that looks like a face, even a particular face.

Believing in miracles

Inspector Gregory: 'Is there any point to which you would wish to draw my attention?' Sherlock Holmes: 'To the curious incident of the dog in the night-time.' Inspector Gregory: 'The dog did nothing in the night-time.' 'That was the curious incident,' remarked Sherlock Holmes.

• Arthur Conan Doyle (from Silver Blaze)

Mary thinks about calling her friend Jill. Suddenly the phone rings and it is Jill.

The coincidence alone does not establish a paranormal cause. Mary may be overlooking all the times Jill didn't call when Mary was thinking about her, or the times someone else called, or the times Jill called but Mary wasn't thinking of her, or the times Jill didn't call when Mary wasn't thinking of her. When Mary thinks about Jill and the phone rings, it registers as an event and something we remember. When Jill doesn't call, it is a non-event. Nothing happened. Nothing is registered and therefore nothing is remembered.

We often pay little or no attention to times when nothing happens. We shouldn't look at past events and find significance in the amazing ones. We need to make comparisons among cases involving no cause or no effect and look at all the other things that might have happened instead.

"The psychic predicted the tornado"

Amazing. It sounds too good to be by chance. What we didn't know was that the psychic predicts a tornado every week. As Marcus Tullius Cicero said: 'For who can shoot all day without striking the target occasionally.' Often we don't notice the incorrect predictions, only the rare moments when something happens. We forget when they are wrong and only remember when they were right. And many times we want them to be right, so we hear what we want to hear and fill in the blanks.

'The art of prophecy is very difficult, especially with respect to the future,' wrote Mark Twain. This is why it's important to remain skeptical of 'future tellers.' Their right guesses are highly publicized but not all their wrong guesses. Like Harvard Professor Theodore Levitt said: 'It's easy to be a prophet. You make twenty-five predictions and the ones that come true are the ones you talk about.' Michel de Montaigne adds: 'Besides, nobody keeps a record of their erroneous prophecies since they are infinite and everyday.'

Some future-tellers place their predictions far enough ahead that they avoid scrutiny when they are wrong. Or they make their forecasts so general they can apply to anyone or to any outcome so they can't be proved wrong.

'There is no evidence that ghosts don't exist.'

Some things can't be proven false. The fact that there is no evidence against ghosts isn't the same as confirming evidence that there are ghosts. What we are justified in believing depends on the evidence, rather than merely the absence of a disproof.

Mary comes home after school and tells John: 'My friend Alice witnessed a miracle.' The 18th Century Scottish Philosopher David Hume suggested a test to analyze claims of miraculous events: 'No testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous than the fact which it endeavors to establish.'

Hume suggests the following test: If the opposite of a given statement is more likely, the statement is probably false. For Alice’s report, the comparison requires considering the testimony’s reliability and alternative explanations, including mistake or deception. Calling an event unexplained does not establish which explanation is correct.

The German poet Johann Wolfgang von Goethe said: 'Mysteries are not necessarily miracles.' That an event can't be explained doesn't mean it is a miracle. No theory can explain everything. As Michael Shermer says, 'My analogy is that the L.A.P.D. [Los Angeles Police Department] can solve, say, 90 percent of annual homicides. Are we to assume that the other 10 percent have supernatural or paranormal causes? No, of course not, because we all understand that the police cannot solve all murder mysteries.'

Bertrand Russell said in A History of Western Philosophy: 'Uncertainty, in the presence of vivid hopes and fears, is painful, but must be endured if we wish to live without the support of comforting fairy tales.'