Jul 29 2019

Coincidence and the Law of Large Numbers

Weird things happen to most people at some point in their lives, and if not to you directly than probably to someone you know. But what is the ultimate meaning to such coincidences? They may seem amazing, and psychologically scream out for an equally amazing explanation.

Skeptics caution, however, that our tendency to see patterns and impose satisfying explanations combine with our relative lack of intuition for statistics to jump to unwarranted conclusions. If we do the math, then it becomes clear that very unlikely events should happen all the time, given enough opportunity.

Some people, however, do not want to give up the narrative value of coincidences so easily. Sharon Hewitt Rawlette, for example, has written a book about The Source and Significance of Coincidences, and prefers a more supernatural explanation. In a recent editorial for Psychology Today she strikes back at the skeptical explanation for coincidences. She basically has one point to make, but first she states the skeptical position:

Skeptics argue that, even if the odds that a particular event would occur at this particular moment to this particular person are very low, there are so many moments over the course of our lives and so many people on this planet, that even very improbable coincidences are bound to happen eventually, just by chance. This is often referred to as the Law of Very Large Numbers.

I give her props for getting this correct, and not resorting to a straw man. That is the argument. She even does some math to support it. She talks about a specific anecdote of a couple rolling an unlikely sequence of numbers (actually every sequence of the same length is equally unlikely, but some occur in more recognizable patterns):

The odds of rolling any particular 20-number sequence on a fair, six-sided die are 1 in 3.7 quadrillion.

We should expect to see chance coincidences of this magnitude about once every 7.4 quadrillion seconds, or once every 3 million lifetimes.

This is where the skeptic appeals to the fact that there are 7 billion people in the world and remarks that, since experiencing something this improbable should happen about once every 3 million lifetimes, we should expect that about 2,000 people on this planet will experience something this staggering just by chance.

Again, yes, sort of.  She still misses the fact that there are numerous opportunities for such unlikely events to occur. She uses an estimate of 2 seconds for the roll, and so concludes that if such an opportunity happened every two seconds throughout one’s lifetime, that would result in one such event every 3 million lifetimes.

In reality this is an impossible calculation to make. First, we can’t really know how many opportunities there are for unlikely coincidences. We would have to consider the interactions among everything that happens in our life and everything else. It’s not just unlikely die rolls, but also the probability of any encounter or event over the course of a day or so having an unlikely connection to any other encounter or event.

Second, you have to consider the odds of all events sufficiently unlikely (as an individual event) to be perceived as a coincidence – not just a 1 in 3.7 quadrillion chance. What about a 1 in a million chance? That will still seem like a coincidence. One 1 in a billion, or trillion – these still count.

Human nature is to seek out and notice patterns. We subconsciously are mining vast amounts of data, looking for those patterns. There are simply too many variables to calculate the odds of any coincidence happening to any person at any time.

But let’s say her estimate is somewhere in the ballpark. This is where she gets to her primary point:

Again, what the skeptic doesn’t acknowledge is that the fact that there are 7 billion people in the world is only relevant if we know how many of those people have or have not experienced a similarly staggering event. The skeptic is making the unfounded assumption that there are not significantly more than 2,000 other people in the world who have experienced something like this—that is, that the number of such experiences doesn’t significantly exceed the baseline that would be expected on chance—but, without data, we can’t make that assumption. There could very well be many more than 2,000 people who have experienced such an improbable confluence of circumstances, and if there were, it would provide strong evidence that something more than chance was at work.

What she is doing here is shifting the burden of proof. Skeptics are simply pointing out that our naive sense of the probability of a coincidence happening is extremely flawed. If you consider the math more thoroughly, you will see that amazing coincidences should happen all the time. That takes a lot of the impact out of coincidences when we experience or hear about them. But if Rawlette wants to claim that coincidences happen more often than we would expect by chance, and therefore a supernatural explanation is required, then the burden of proof is on her to provide the data. Otherwise she is also making an argument from ignorance.

We also have to recognize that human memory is flawed. Because of confirmation bias, we tend to help coincidences happen by subconsciously tweaking our memories to bring things into better alignment. Our brains prefer the theme of a memory over the details, and can seamlessly alter those details to enhance the theme, such as the interest in an apparent amazing coincidence.

There is a situation, however, where we can do some specific statistics – where we are examining a more limited claim that allows for the control of some variables – how often should we expect someone to win the lottery twice. Winning the lottery once can be millions to one odds, so twice must be millions times millions, or quadrillions to one against. That claim is so common the basic logical fallacy is named after this example – the lottery fallacy.

This comes from asking the wrong question – what are the odds of John Smith winning this lottery twice? There may be other assumptions in there – buying only one ticket on two occasions. So first we have to ask how many tickets does the average lottery player buy, how many lotteries do they play, and over what period of their lives? Then we have to consider all the people playing the lottery. So the real question is – what are the odds of anyone winning any lottery twice?

Here is a calculation of anyone winning a specific lottery twice, showing that even though each ticket has a 1 in 14 million chance of winning, the probability of anyone winning twice over a period of 20 years is about 60%. In fact, for many lotteries someone should win twice every couple of years, and that is what we observe. There is no statistical excess of double lottery winners.

Sports is another arena where were have thorough statistics. In baseball, for example, statisticians have looked at the frequency of perfect games (there have been 18 in the history of the major league). This is in accordance with models, which includes the fact that perfect games are more likely when above average pitchers go up against below average hitting teams.

Whenever we have a finite system where we can do the math, it checks out, with no need to invoke supernatural coincidences or other factors. But we just can’t do the math with the open-ended question of how often we should expect an undefined coincidence of undefined improbability, given an undefined number of potential opportunities. We can say, however, that they are more likely than common naive assumptions and lack of facility with statistics would lead most people to believe.

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