In my recent post, "Wittgensteins Everywhere," I noted the coincidence that I ran across a couple of references to Prince Wittgenstein, in two different books, on different topics, on the same day. I adumbrated (facetiously, I assure you) the possibility that there was some "Grand Plan" causing these apparently coincidental events in my life. A friend has since cautioned me about attributing events to "grand plans."
That's always a good caution. After all, as is well known, incompetence (greed, normal human foibles) is a better explanation of untoward events than conspiracy. (Yes, I think Lee Harvey Oswald acted alone. And, while I found Kevin Phillips' book American Theocracy interesting, I don't for a moment believe that there is a conscious, intentional, conspiracy underlying the events Phillips finds to be so suspicious.)
Moreover, if one has a sufficiently large sample, one will often find "strange coincidences" that are not strange at all. One of my favorites in the realm of statistics is that there was a perfect correlation between the price of rum and the salaries of Congregational ministers over quite a long period of time. Was there a secret compact between the rum dealers and the Congregational Church (which was, after all, the established church in Massachusetts)?
Not really. Over a very long period of time, prices have trended upwards. This is due to a number of things: rising populations, exhaustion of some resources, a constant tendency of governments to debase the currency, the discovery of gold and silver in the New World, and so on. Since prices of most goods and services are rising most of the time, all prices are positively correlated to some extents. (This is how your stockbroker can assure you that the stock market always, in the long run, rises. However, as Lord Keynes once said, in the long run we're all dead.) If one has a sufficiently large universe of prices, two of them will be perfectly correlated.
There is an old trick question illustrating this phenomenon. How many people do you have to have in a group before the chances of two of them sharing the same birthday go above 50%? Well, you might reason, there are 365 (and 1/4) days to choose from, so there would need to be 183 people in room before one of them has the same birthday I have. True, but that wasn't the question. The correct answer is, as I recall, 24. (Corrections and explanations are welcome.)
So, it doesn't really take all that many cases to have a sufficient universe for various "coincidences" to appear. If I ask, "Isn't it strange that two very different books happen to mention the same man?" - that looks like a coincidence. If, on the other other hand, I ask, "Out of the hundreds of men mentioned in the dozens of books I've read this year, wouldn't it be odd if some of them weren't mention in more than one book?" - then I'm looking at a predictable regularity.
Part of the fortune-teller's bit is to throw out so many vague predictions that some of them are bound to agree with some future event. But even more of it is due to our constant need to create patterns. When the patterns really do exist, as in the natural selection of biological traits, this human propensity is quite useful. When, however, the pattern isn't really there, one ends up with a specious theory. Astrology, for example, isn't just wrong because the stargazers can't agree on which predictions to make for which combinations of signs. Rather, it's wrong because it is based upon the positions of the constellations in the heavens. And the constellations do not exist. The stars really are randomly distributed, with respect to their visibility to a viewer on the Earth, so that any so-called "constellation" is purely a creation of the observer. Since the constellations don't correspond to any real pattern, they can't be predictive of events.
Unless, of course, you want to contend that the particular constellations we "see" are a result of some hard-wired characteristics of our minds - characteristics which also determine how we react to various circumstances. Sort of a celestial Rohrschach test.
Glenn A Knight
In my study
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2 comments:
> How many people do you have to
> have in a group before the
> chances of two of them sharing
> the same birthday go above 50%?
> ...The correct answer is, as I
> recall, 24. (Corrections and
> explanations are welcome.)
It's actually 23 people. It seems unlikely, but it's true, and it's pretty easy to see why it must be so.
One of two mutually exclusive events MUST occur: At least two people in the group share a birthday, or no one in the group shares a birthday. If the group is 2 people, the probability they DO NOT share a birthday is 364/365 = 0.997; thus the probability they DO share a birthday is 1 - 0.0997 = 0.003, or 0.3%.
However, every time another person is added to the group, another independent event is added to the probabilities; the probability of independent events co-occurring is the product of their individual probabilities.
So, when we add person #3 to the group, the probability that he will NOT share a birthday with one of the other two is 363/365 = 0.995. To determine the probability of the two independent events (#1 and #2 not sharing a birthday AND 3# not sharing a birthday with either of them) co-occurring, we multiply 0.997 X 0.995 = 0.992. Thus, the probability that any two of the three DO share a birthday is 1 - 0.992 = 0.008, or 0.8%. That's a very small probability, but note that it's more than double the probability when there were 2 people in the group.
As we add more people to the group, the probability of at least two members sharing a birthday increases much more quickly than seems possible. When we get to 23 people, the probability is 0.507, or 50.7%.
For 30 people, the probability is 0.706, or 70.6%. That's when you should bet big money if anyone in the bar is impaired enough to go for it.
Most of what seems unlikely, random, coincidental, or even eerie in our lives is actually the result of the workings of well-understood statistical principles.
Thank you, Lloyd. I should have known that I could count on you to come up with the answer to a statistical question.
On the subject of coincidence, another way to get generate them is to re-use your data. This is a no-no in real science, and here is a story to illustrate that:
"Once, during a public lecture, [Feynmann] was trying to explain why one must not verify an idea using the same data that suggested the idea in the first place. Seeming to wander off the subject, Feynmann began talking about license plates. 'You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won't believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the stae, what was the chance that I would see that particular one tonight? Amazing."
Another "coincidence" in which I am personally involved, is that my son's parents-in-law were married the same day my wife and I were. That would be, I think, a variation on the birthday question. That is, the odds that any two couples in the room would share the same wedding anniversary would be the same as the odds that any two individuals would have the same birthday. The year would add a dimension to it, but, when you consider that, having kids of an age to marry each other, we were likely to have married within a few years. (There are exceptions to that. My brother is essentially of a different generation than that to which his wife belongs: his in-laws are my age, rather than the age of our parents.)
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