Pitfalls in Elementary ProbabilityAuthor(s): Samuel A. GoudsmitSource: Proceedings of the American Philosophical Society, Vol. 121, No. 2 (Apr. 29, 1977), pp.188-189Published by: American Philosophical SocietyStable URL: http://www.jstor.org/stable/986527 .

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PITFALLS IN ELEMENTARY PROBABILITY

SAMUEL A. GOUDSMIT

Professor of Physics, University of Nevada

(Based on a lecture presented November 11, 1976)

WE USE probability considerations without being aware of it in many daily-life situations. When- ever we make a choice or take a risk, we subcon- sciously decide that the probabilities are in our favor. We do this by intuition and I wish to point out that intuition can be a dangerously mis- leading guide in probability matters. I shall show this by discussing a few examples.

Let me start with the example of tossing a coin. Everyone knows that with an honest coin the chances for throwing heads or tails are equal. The reasoning is simple, it has to be one or the other and we see no grounds for favoring either one, thus the odds are fifty-fifty. Well, it is not so. I learned this many years ago from an under- graduate student at the University of Michigan. He lived in Brooklyn and earned his tuition by working summers at Coney Island. There he learned things probably of greater use to him than what we taught him at Michigan. He showed me how to toss a coin and make it come down heads or tails at will. I never had the dexterity to learn the trick, but some of my Michigan students became quite proficient at it. It is most easily done with half-dollars. The point is that the equal chance holds only if the coin is tossed so that it rotates freely about an axis in the plane of the coin. My Michigan stud- ent tossed it so that it wobbled violently, but always kept the same side up in flight; it never flopped over. He caught it on the back of his hand, knowing in advance which side was up. He easily managed to get "heads" a dozen times in a row. Thus the odds depend on the movement of the coin; one must examine in detail the mathematics of the motion of a disc. Even if the tossing is done by a machine, the motion may have a component of rotation about an axis per- pendicular to the coin. This causes a bias in the results. I heard of a similar mechanism for throwing dice, but it is not as foolproof as the coin-tossing case. There are supposed to be experts who can throw dice with just the right amount of push so that they know how many times

the dice will turn over before they stop. These experts can often, although not always guess the outcome of the throw.

A second example was first reported by the American Philosophical Society. In 1938 Frank Benford published a paper in our Proceedings ' entitled "The Law of Anomalous Numbers." He had made a very unusual discovery. He examined tables of numbers of the kind found in the World Almanac, such as the population of the fifty states. The entries start with the digits 1, 2, 3, up to 9. Our intuition sees no reason why any value of this first digit should be preferred over any other. Benford found that this was not in agreement with the facts. The larger values occur less often than the smaller. If we count all the entries start- ing with 1, 2, and 3, we find that they occur a little more often than all of those beginning with 4, 5, 6, 7, 8, 9. Our intuition would have pre- dicted twice as many of the latter.

The article was followed by an important physics paper by our fellow member Hans Bethe and M. Rose. That is the reason why the Ben- ford article caught the attention of physicists. If probability calculations are properly applied, the Benford results are no longer "anomalous." This was pointed out for the first time in a short note which I published 2 in 1945 in Nature together with Wendell Furry from Harvard, where I was visiting that year. Since our note appeared, about forty papers have been published on this "first digit problem," extending and refining our short derivation. The arguments are rather involved. The simplest explanation of the Benford phe- nomenon can be found in that delightful paperback Lady Luck by our fellow member Warren Weaver.3

1 Frank Benford, "The Law of Anomalous Numbers," Proc. Amer. Philos. Soc. 78 (1938): p. 551.

2 S. A. Goudsmit, and W. H. Furry, Nature, 155 (1945) : p. 52.

3Warren Weaver, Lady Luck (Science Study Series, Anchor. Books, New York, Doubleday & Co., 1963).

PROCEEDINGS OF THE AMERICAN PHILOSOPHICAL SOCIETY, VOL. 121, NO. 2, APRIL 1977

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VOL. 121, NO. 2, 19771 PITFALLS IN ELEMENTARY PROBABILITY 189

The third example comes from navigation. In most systems of navigation the navigator, by ob- serving the stars, a lighthouse, or a radio beam, determines a "line of position" on his map of the area. His location is a point somewhere on that line. By making a second observation, he obtains two intersecting lines of position and his location is thus at the point of intersection. He checks this by a third observation. In an ideal case the three lines should intersect at the same point. However, errors in observation cause deviations and the three lines form a little triangle near the point where they should meet. Thus the navigator's intuition tells him that his true position is somewhere inside that little triangle. It is true that the most probable location is inside the triangle, but only in one out of four cases will the actual location be inside.4 This rather disappointing and surprising result can easily be proven by those familiar with elementary prob- ability calculus. The only assumption involved in this result is that for each line of position errors to the right have the same probability as errors to the teft. But these error probabilities can be different for the different lines. One may be very accurate, another may be a crude measurement.

The fourth of my examples can best be demon- strated with a deck of playing cards. I let some- one in the audience pull three cards out of a well- shuffled deck. Showing the three cards to the audience I can truly state that "this combination of three cards is obtained on the average only once in about 20,000 draws." But then I add, "this person obtained this combination on the first draw, thus you just witnessed a miracle." Every- one laughs at this, knowing quite well that it is not a miracle. However, it is a risky demonstra- tion, because some day the person in the audience may draw three aces, or three kings, or an ace, king, queen of the same suit, or any one of many such special combinations. The chance for this to happen is not so small. When this happens the impact of my remark is lost, because we really think that we observe a miracle. There is of course, no difference in the odds, no matter what cards are drawn. When three kings are drawn this is just a coincidence. Unfortunately many people wrongly give special significance to such coincidences. Arthur Koestler wrote a book about them. Edgar Allen Poe warned in The Murders in the Rue Morgue that "coincidences in general,

4S. A. Goudsmit, Navigation 1 (1946): p. 34.

are great stumbling blocks in the way of that class of thinkers who have been educated to know nothing of the theory of probabilities."

An eminent physicist and friend of mine errone- ously believed to have proven a phenomenon be- cause coincidences he observed would normally occur only once in a hundred years. His wish- ful thinking made him forget the rules of elemen- tary probability. Much worse was that it required a lot of effort on my part to convince some promi- nent colleagues that my friend's conclusions were unwarranted. These colleagues too had hoped that the results would be right. Coincidences also play a significant role in the pseudo-scientific argu- ments in favor of parapsychological phenomena.

The next example can be illustrated best by an anecdote from one of the books by the great sailor Captain Marryat. These books were still avidly read by boys when I was a child, though they were then already about eighty years old. A midshipman asked a more experienced com- panion what to do when the wooden ship was being bombarded by the enemy. The answer was that he should place himself in front of the hole made by one of the first cannon balls. The mathe- matics professor at the naval academy had com- puted that the chance of two cannon balls passing through the same hole was negligibly small, thus he would be perfectly safe there! Though we know that this is nonsense, we nevertheless do make this mistake from time to time. For ex- ample when "red" has come up at the roulette wheel six or seven times in a row, there are many among us who believe in a "law of averages" and expect "black" to appear soon and place their bets accordingly. This is of course wrong. The little ball jumping around the wheel has no mem- ory, it does not know what happened before. Nevertheless, it is hard to rid oneself of this erroneous reasoning while gambling.

These few examples are, I think, sufficient to warn us about the dangers of following our intui- tion in matters of risk. But like all warnings, it won't do much good. I have observed eminent mathematicians who at the gambling tables seemed to have forgotten all they knew about probability. This should not be construed as a warning against gambling. In fact some games such as state lot- teries can make good economic sense even when the mathematical odds are against the gambler. The ratio of the economic values of winnings and stakes can be much larger than the ratio of their monetary values.

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