… as long as you don’t specify beforehand what miracle you might be interested in observing.
A very good friend of mine, nomasir, had an interesting post recently on his blog, referring to a British news article about a grandfather, father, and son who were all three born on the same day of the year. The article suggested that this was a very unlikely occurrence.
Just how unlikely was the matter of debate in the blog post… and that discussion reminded me of an interesting article by Persi Diaconis, “Statistical Problems in ESP Research” (Science, July 14, 1978, 201(4351):131-136). Diaconis has contributed a lot of interesting mathematics, with applications ranging from magic tricks to card shuffling to coin flipping to, in this case, evaluating– and usually debunking– claims of ESP (extra-sensory perception).
I had read the article before, but in the process of searching for it, I also stumbled across something I had not seen before: a series of letters in response to the article by several apparently less-than-impressed researchers in the field, along with a response from Diaconis. I recommend both the article and the letters; it’s an interesting exchange, short and readable even for the non-mathematician.
My point of this post deals with what Diaconis refers to in the article as the “problem of multiple end points.” (The relevant section of the article involves the experimental subject referred to as B.D.) In the context of the coincidental birthdays mentioned above, we can calculate the (small) probability of that particular event… but is that probability really interesting or useful after the event has occurred, when we didn’t “bet on” the event before it occurred?
Suppose that, instead of three generations of men all having the same birthday, they had all died on the same day of the year. Or maybe it was a grandmother instead of a grandfather, or daughter instead of a son, etc. Or maybe the three birthdays were all within just one or two days of each other. These would all still be “interesting” occurrences. Now– what is the probability that some one (or more) of these various “interesting” events might occur?
Quoting Diaconis again, “the odds against a coincidence of some sort are dramatically less than those against any prespecified particular one of them.”
I am reminded of the story of David Wilkerson, a man who claimed to have been inspired to make hundreds of peanut butter sandwiches on the night of 10 September 2001, which were then fed to rescue workers and victims after the terrorist attacks, the idea being that such inspiration was somehow a prediction of the attacks. (This claim has since been discredited, but that’s not the point, so let’s suppose for example that it did actually happen.) As one news story at the time described it, “What were [the sandwiches] for? Who would eat them? That part wasn’t clear [my emphasis]…”
Now suppose that tonight you stay up all night making peanut butter sandwiches. What is the probability that you will not find some good use for them, even if it’s not in response to an unnatural disaster with global impact?