PETRTSON-MIDDLEBURY (Winter, 1993)
I introduces coincidences by having students describe an ``amazing coincidence" that they have experienced. (Tom Moore has also used this as an early weiting assignment in his seminar course at Grinnell). I was going to initiate a discussion of the degree to which people found other's stories amazing, but didn't want any one to feel ridiculed in any way for sharing a story, so I backed off on the plan (Does anyone have any ideas here? The point is that we tend to find our own coincidences much more striking than those of other people. On some level we are apparently willing to accept that coincidences are bound to happen somewhere to someone, but are quite incredulous when they happen to us. See Falk, 1989 in references for a formal psychological study on just this point).
I next introduced the venerable birthday problem. This deviates from current news strategy, but I can add some local flavor: of 10 mathematics faculty at Middlebury, two share a birthday (there are also two unrelated Peterson's). It also figures prominently in the Diaconis-Mosteller analyses. The birthday problem, of course, asks how many randomly chosen people must be selected before there is a better than even chance of at least one pair of matching birthdays (answer: 23). After solving this I introduce the "birthmate" problem, which asks how many people you must randomly select to have a better than even chance of matching your own birthday (answer: 253). The Paulos book Innumeracy gives an amusing example of the confusion between these two, from an episode of the Tonight Show. It seems a somewhat mathematically informed guest was attempting to explain the birthday problem to Johnny Carson, who could not believe the answer of 23. He polled the studio audience of 200 people, looking for a match for his own birthday, and found none. The guest, unfortunately was at a loss to explain the distinction. I found this to be a good vehicle for discussion the human tendency to personalize coincidences, getting at the distinction between "how likely is an event like this to happen" vs. "how likely is an event like this to happen to me." Chapter 2 of Paulos is devoted to probability and coincidence, and gives a lot of discussion to the tendency of "innumerates" to personalize in their assessment of probability of a coincidence, reasoning in terms a specific case (their own birthday ) rather than the general case (some two people's birthday's match).
We then turned attention to the following analysis of the ``365 New Words a Year Calendar 1991", which is reproduced from the March 1992 Harper's Magazine (v 284, n 1702. p.19).
The initial reaction was the kind of surprise that I'm sure the publishers had in mind! There followed a good discussion of what might be going on. What follows is a summary of ideas. First, it was observed that we should be even more surprised if none of the words on a current vocabulary building list turned up in news reports over the year (In fact, Diaconis and Mosteller include a model for running across a word we have recently learned). It was observed that ``maelstrom" and ``palpable" are not really specific to the news stories. As for the daily correspondences, it was observed that 1/17 was the second day of the war. How close would it have to be to the first day? would any time during the war do? how about the start of the ground offensive? (This is what Diaconis and Mosteller refer to as multiple endpoints or ``almost birthdays."). Similarly, coup appears in a definition on the day Gorbachev returned to Moscow, not on the day of the coup itself.
I found this list somewhat reminiscent of the famous list of Kennedy- Lincoln assassination correspondences, which most of the students recalled having seen when I mentioned it. (I am tempted to try giving an experimental assignment where students have to construct list like the above on their own!) This discussion above leads in to the "blade of grass on the fairway" principle. I have read descriptions of this in reports on the recent Diaconis-Mosteller work, but I believe this name has been around quite a while. As a golfer prepares to hit his first shot off the tee. the probability that his ball will come to rest on any particular blade of grass on the is quite small. Nevertheless, it will come to rest on some blade. So we know a priori we are about to observe s o m e event of very low probability. Therefore. surprise expressed after the fact about the particular blade hit is not warranted.
Rounding out the blade of grass idea, somewhat tongue-in-cheek, I had a writing assignment base on two slightly dated articles I had been saving fromThe Boston Globe. They concern the 1989 U.S. Open golf tournament, where in less than two hours, four golfers, all using 7-irons, scored holes-in-one on the sixth hole of the course. The first article is full of the typical effusive prose that characterizes popular reports of coincidences:
The odds? A dissertation may be needed in the mathematical journals. but it seems the odds may be I in 10 million (from a University of Rochester mathematics professor), to I in 1,890,000,000,000,000 (that's quadrillion, according to a Harvard mathematics professor), to 1 in 8.7 million (according to the National Hole-in-One Foundation of Dallas), to I in 332,000, according to a Golf Digest calculator, who added that, statistically, this will not happen again in 190 years.
In the second article, the author calms down and consults a mathematician from U. Mass. on how to reconcile the many figures presented. For their assignment, I had the students hypothesize what the 3708-to-one odds quoted for a hole-in-one might represent, and then reproduce the calculations cited for various alternative events (e.g., four golfers, any given day of play, in any tournament during the year's pro tour).
This seems to have been a very successful unit of the course. One of my students even produced a news article describing how Japanese golfers, who by custom are required to throw lavish celebrations for their playing partners and business associates should they themselves score a hole-in-one, actually take out insurance policies to cover the expenses (see references) .
PETERSON-MIDDLEBURY (Fall, 1991)
I didn't do Coincidences as a topic in this course, but a number of the ideas came up when we treated Lotteries. Here the story of Margo Adams. who won the New Jersey Lottery twice in four months, provided interesting discussion on two issues.
First, the distinction (noted above in the context of birthdays) between this happening to someone in the U.S. (given the number of states lotteries operating and the large number of people who play them) as opposed to its happening to this particular individual in this particular state, on these particular tickets, etc. The staggering odds (17 trillion to one against) reported in the popular press are in fact correct for a particular person buying exactly two ticket for the NJ lottery and winning with both.
There is the related fact that Ms. Adams herself greatly increased her purchases of lottery tickets after her first win (buying several hundred a week thereafter!) was by and large ignored in the news reports about her second win. So it is with many "amazing" outcomes. We forget the numerous instances of failures, and are startled by the amazing successes-even if the successes are occurring at no more the rate that would be predicted by probability theory.
SOURCES
News Stories
Baskin, Anita. Presidents on parallel planes. (demystifying coincidences between the deaths of Abraham Lincoln and John F. Kennedy) Omni Jan 1993, v 15, n4, p79( 1).
Deans, Bob. Japan's golfers insure against success. Atlanta Journal, August 11, 1991, pB4(1).
Harvey, Lucy. Just a coincidence? (from an analysis of the '365 New Words a Year Calendar'). Harper's Magazine, March 1992, v 284, n l702, pl9(1)
Michael Madden. At the US Open, aces wild on sixth. The Boston Globe, June 17, 1989, pl (Sports).
Michael Madden. Doing his odds job. The Boston Globe, June 27, 1989, p39.
Discussion articles
Blackmore, Susan. The lure of the paranormal. New Scientist Sept 22 1990, v l 27, n 1735, p62(4).
Hanley, James A. (1992). Jumping to coincidences: defying odds in the realm of the preposterous. The American Statistician 46(3), 197-202.
Discussion of flawed analyses in news reports. of lottery and birthday coincidences.
Gould, Stephen J. (1989) The streak of streaks. Chance 2(2), 10-16.
Title refers to Joe DiMaggio's 56-game hitting streak in the 1941 baseball season. Eloquent discussion of the apparent human need to find transcendent meaning in our percepetions of pattern.
Mock, Carol and Weisberg, Herbert F ( 1992). Political innumeracy: encounters with coincidence, improbability, and chance. American Journal of Political Science 36. 1023-1046.
Picks up on ideas from the Diaconis Mosteller article. Analysis of success of presidential administrations as "predicted" by astrological data. Discussion of the pitfall of searching for significance through multiple tests of hypotheses .
Paulson, Richard A.. ( 1992). Using lottery games to illustrate statistical concepts and abuses. The American Statistician 46(3), 202-204.
Technical Articles
Falk, Ruma (1989). Judgment of coincidences: mine versus yours. American Journal of Psychology , v 102, n4, p477-483.
Includes: Comparison of surprisingness ratings of self-stories and other stories. (table); Comparison bet. mean surpnsingness ratings of self &other coincidences.
Diaconis, Persi and Mosteller, Frederick (1989). Methods for studying coincidences. Journal of the American Stalistical Association 84, 853-861 .
General References
Kahneman, D, Slovic, P. and Tversky, A. (Eds.) Judgement Under Uncertainty: Heuristics and Biases. Cambridge University Press. 1982.
Feller, William. An Introduction To Probability Theory and its Applications.
Paulos, John A. Innumeracy: Mathematical Illiteracy and its Consequences. New York: Hill and Wang, 1988.