At the August 2001 meeting of skeptics in Toronto, I carried out an experiment to illustrate how poorly people estimate the chances of coincidences occurring. I asked people entering the meeting to give their birth dates in months and days, like August 12 or October 31.

Having collected 64 such dates, I asked what the odds were of two people among those 64 having the same birth date. Given 365 days in the year, the most common estimates given by audience members were 64 out of 365, roughly one in six, although some people said one in three or similar odds.

Would you believe the mathematical odds were actually 997 out of 1,000? In other words, it was almost a dead certainty we’d have at least one pair of matches. And, in fact, it turned out we had two pairs of people at the meeting with the same birth dates.

(In case you think Toronto audiences are particularly strange, I did a similar experiment two months later at the Sudbury chapter meeting with 34 birth dates. Again no one guessed the odds were greatly in favour of finding matches, and there we found three pairs with the same birth dates!)

Also at the Toronto meeting, I took the phone numbers of 200 people from the OSSCI database and asked the meeting what it thought the odds were of finding phone numbers with the same last three digits. Since there are 999 possible combinations of digits, some people figured the chances of a match were 200 out of 999, or roughly one in five, although some picked better odds.

Everyone, including I, was amazed however to learn that 31 of the 200 phone numbers shared the same last three digits as other phone numbers in the sample.

How could the chances of coincidence be so great in these instances and how could we be so far off in estimating those odds?

Confirmed by virtual reality

The chances of at least two people in any group sharing a birth date can be presented mathematically in quite a complicated formula, pointed out to me after the meeting by OSSCI Secretary-Treasurer Michael De Robertis. You can find several variations of this formula on the Internet. But in the case of 64 people, it yields a figure of about .997 — 1.000 being 100 percent certainty.

Here I had been worried we would not have a single match for my little experiment at the meeting, and now I was being told we had such overwhelming odds on our side! It didn’t seem possible. Odds of 997 out of 1,000 seemed too far from intuitions for even me to accept.

So, being an obsessive skeptic, I created a computer program that generated random birth dates to check this out. It generated 1,000 sets of 64 dates chosen at random and, lo and behold, 996 of the sets contained matches. On a second 1,000 sets it got 998 containing matches. Similar runs got similar results. The calculated probability of .997 was confirmed by virtual reality, though it still seemed amazing to me.

Complex formulas are hard for us non-mathematicians to understand. To try to get a more intuitive grasp of how the odds could be so great, look at it this way.

Take Susan sitting in the audience. The odds of Susan finding that the stranger sitting beside her has the same birth date as she is obviously one in 365, since there are 365 days in a year (ignoring leap years).

But in a meeting with 63 other people — with 63 other birth dates to compare to Susan’s — the odds of one of them matching her birth date are 63 times greater, that is, 63 out of 365, roughly one in six. This is where most of us stop figuring and come up with our rough estimate, which turns out to be far too low.

We forget that each of the other 63 people in the room has the same 63/365 — or roughly one in six — chance of sharing a birth date with someone else. When we add up all these possibilities, we see the odds of at least one of the 64 people sharing a  birth date with someone else in the room is 64 times greater than Susan’s odds alone. Now we can see that it is almost a sure thing that at least one pair can be found.

The precise calculation is a little more complicated, but this gives you an inkling as to how the odds could be so much higher than one might think at first.

Why are we so wrong?

The problem is that when most people are asked to figure out odds, they don’t do the math past the initial step. They might vaguely think, “Well, there are 64 people here so I guess my odds are about one in 64.” Or they figure, “Sixty-four people have a possible 365 birth dates, so that’s about one in six.”

It is quite natural to do this limited kind of rule-of-thumb estimating. During the evolution of our brains, our survival did not likely depend very often on figuring odds with any deeper complexity, so we have not developed an easy facility for complicated calculations. It requires a determined effort to go beyond rough guesses.

We also tend to be egocentric – to calculate odds only as they affect us, or affect just one person – so we seldom get past figuring out, for example, our own odds of sharing a birth date with someone in the room. We fail to go on to consider the combined odds for everyone.

Suppose I ask you the odds of Roger Nesmith of Mississauga winning the lottery last month, given that 1 million tickets were sold. You might say one in a million, and you’d be right.

Suppose I were to reveal then that Roger Nesmith of Mississauga did in fact win the lottery last month. Since the odds were one in a million, shouldn’t his winning — overcoming such great odds — qualify as a miracle?

No, because somebody had to win the lottery, you say. Looking beyond just one person’s odds, you realize that it’s nearly certain that one of the million tickets would win. However, miraculous it may seem to Roger Nesmith, it is not significant when one considers the overall odds.

Yet it always seems like an incredible miracle when something with seemingly very long odds happens to an individual, particularly to us, but taking into account the entire field of odds for everyone combined puts it back into perspective.

What has this to do with skepticism?

Many times I have heard from psychics and believers in the paranormal that the odds of such-and-such a thing occurring through natural means were astronomically long, and therefore supernatural forces must have been at work.

If you hear this argument, try to think about what the entire field of opportunities for such occurrences would be. Be aware that coincidences happen all the time and only seem significant because people do not always understand the true odds of those events occurring at the same time or place, given a world of zillions of events happening. Remember that the initial feeling that long odds are involved is often grossly mistaken.

And remember the coincidences of birth dates in a room full of skeptics.