How Skeptics Misapply the Law of Very Large Numbers

Sharon Hewitt Rawlette

Source: courtesy of Sharon Hewitt Rawlette

Sharon Hewitt Rawlette has written an excellent compendium of coincidence stories with explanations and uses, The Source and Significance of Coincidences. In this guest editorial, she takes apart the Law of Very Large Numbers.

A world population of 7 billion people doesn’t automatically make your coincidence experience insignificant.

When we experience a strange coincidence, we often remark to ourselves, “What are the odds of that?” If we’re feeling ambitious, we might even try to calculate the odds that such a bizarre confluence of circumstances would happen just by chance. After all, the lower the odds of a coincidence’s happening just by chance, the greater the odds that the coincidence is a sign of some deeper process at work—perhaps even something psychic or paranormal.

However, there’s an argument frequently made against the idea of coincidences’ being significant, statistically speaking. 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.

But does the Law of Very Large Numbers really mean that chance is the best explanation for even the most unlikely of coincidences? It doesn’t, and here’s why. The existence of 7 billion people on our planet is only relevant to our coincidence experiences if we know how many of those 7 billion people have or have not experienced coincidences as striking as the ones we’re considering. That is, large numbers are only relevant if we have data for those large numbers.

Let me explain by way of an example.

In my recent book, The Source and Significance of Coincidences, I discuss an event that happened to Dr. Andrew Paquette, which he wrote about in his memoir Dreamer. Paquette was playing backgammon with his wife when he suddenly had the intuitive conviction that they were about to roll matching descending doubles, from six to one. He announced this conviction to his wife, and they then proceeded to roll the dice. His wife rolled double sixes; then he rolled double sixes. She rolled double fives; then he rolled double fives. The pattern continued until they’d both rolled double twos. At that point, Paquette was aghast at the extraordinary improbability of what had just happened. He began to doubt that they would be able to roll the clinching pairs of ones, and he told his wife as much. The next two dice then showed non-matching numbers.

Although Paquette and his wife didn’t roll matching numbers all the way down, they still rolled the first 20 numbers of the sequence Paquette had predicted. The odds of rolling any particular 20-number sequence on a fair, six-sided die are 1 in 3.7 quadrillion. So it was pretty unlikely that Paquette and his wife would make that particular series of rolls at that particular time.

Nevertheless, it’s true that there are many, many opportunities for coincidences to arise over the course of our lives. So we have to ask: How often should we expect such an improbably significant event to happen to a person just by chance? Once a week? Once a year? Once in a lifetime? Once in every ten lifetimes? We should expect to see chance coincidences of this magnitude about once every 7.4 quadrillion seconds, or once every 3 million lifetimes. (The details of the calculation have to do with how long it takes to observe the event—I conservatively estimate that it would take at least two seconds—and how many other rolls of the dice were equally likely on chance: 3.7 quadrillion.)

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—and thus Paquette’s experience is no evidence for the existence of psychic powers or anything else paranormal.

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.

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Now, the skeptic is probably thinking that if significantly more than 2,000 people in the world had had something like this happen to them, we would have heard about it on the news. But is that true? If it would happen on chance to 1 in 3 million people over the course of their lives, that’s only 100 people in the whole United States. The actual rate of these experiences could be double or triple that rate, with 200 to 300 people in the US having this experience sometime over the course of their lives, and it could still easily fly under the media radar, especially considering that many of those who experience such things don’t reveal them even to their closest friends and family, for fear of ridicule.

People’s reluctance to talk about such experiences means that when you reflect on the implications of one of your own unusual experiences, it’s important to think about how large your reliable data set actually is. Rather than ask yourself, “How likely is it that this event would happen to someone on a planet of 7 billion people?” ask yourself, “How likely is it that this event would happen to me or to one of the people I know who would confide in me if it happened to them?” Let's say you have a circle of only 10 friends whom you know would have told you about this kind of experience if they’d had it. And one of you has actually had an experience that should only be expected on chance to happen to one in 100 or one in 1000 people. That’s strong evidence that that experience was not just chance, that there was indeed something deeper at work.

As for Paquette, given that he publicized his experience, he probably has a much wider circle of people who would have told him if they had had experiences on the same level of improbability as his own. However, even if we estimate the size of this circle at 50,000 people (and conservatively assume that none of them has had any equally improbable experiences), that still makes the occurrence rate in Paquette’s sample 1 in 50,000—60 times higher than the expected rate of 1 in 3 million. This means that Paquette’s experience gives him very strong reason to believe that something besides chance was going on the day he and his wife rolled matching descending doubles. And the fact that there were 7 billion other people on the planet doesn’t matter at all.

Make no mistake: There is truth to the Law of Very Large Numbers, but it can only be properly applied when we have data for those large numbers.