Nahin Paul J. - Duelling Idiots and Other Probability Puzzlers

Author : Nahin Paul J.
Title : Duelling Idiots and Other Probability Puzzlers
Year : 2000

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ln 1965, after spending nearly three years as a designer of digital machines in the Systems Division of Beckman Instruments, Inc., in Fullerton, California, my employment suddenly ceased. The sad reason for this was euphemistically called by upper management a "change of business opportunities." That is, the digital machinery product line was immediately terminated for a lack of any new customers and I was, at the age of twenty-five, looking for a new job. Fortunately for me, Hughes Aircraft Company had its Ground Systems Group (the division that made groundand ship-based radars) in the same town, and its management was hiring young electrical engineers. So in December 1965, I became a member of the technical staff at Hughes-where I quickly learned that Boolean algebra and sequential switching theory, which had held me in good stead as a digital systems designer at Beckman, simply wasn't going to be enough for long-term survival as a radar systems analyst at Hughes. I needed to learn sorne more math. I needed to learn probability theory, and I needed to learn it fast. As astonishing as it is now to look back on those days, in 1962 I had graduated from Stanford University-one of the world's great schools-with a B.S. degree in electrical engineering without having taken a single course in probability theory. lt wasn't that I was lazy, as no one in my class of electrical engineers (EEs) took such a course. lt simply wasn't required, and my advisor had never brought it up, even as a suggestion, because everybody thought of probability theory as graduate-level course work. And when, in 1963, I graduated with my master's degree from Caltech, which most people consider to be a veritable hothouse of techno-nerds, it had been neither required nor suggested that a first-year graduate student in electrical engineering study probability. lt was only when I started my doctoral studies in electrical engineering at the lrvine campus of the University of Califomia as a Howard Hughes Staff Doctoral Fellow in 1968 that I took a formai probability course in a degree program. But by then I had been at Hughes for nearly three years and had already started such studies myself, for the most practical of reasons: in order to keep my job. Actually, even while still at Beckman, I had been exposed to a famous probability question, although I hadn't recognized it as such at the time. lt was a twice-a-day routine for groups of the engineering staff to take what was jokingly called a "roach-coach" break. That is, several of us {let's say N people) would, moming and aftemoon, take ten minutes to walk out to the parking lot and buy donuts and coffee from the visiting lunch van. Rather than each of us individually paying for our own purchases, however, we played a game called "odd-man-out": each of us would simultaneously flip a coin and then show all the others what we had gotten. If it tumed out that everybody but one had the same result (N- 1 heads and one tail, or N- 1 tails and one head) then the "odd" person paid for everyone. Otherwise we all flipped again, and so on until we got an odd man out. There are several interesting questions about this game, but one of immediate practical interest concerns how long it will take, as a function of N, to get an odd man out. That is, how many flipping attempts (on average) will it take to reach a decision on who pays? And what if one of the coins is biased? You will see in Problem 9 that it is actually quite easy to calculate the answers once we have established sorne fondamental theoretical results. I played "odd-man-out" from 1963 to 1965, all through my stay at Beckman (Hughes was a much more formai place, and I never saw anybody play for donuts and coffee during my six years there), but it never even occurred to me that such questions had answers. And if it had I wouldn't have known how to find them. Today, of course, such an admission would be considered tragic. I presently teach sophomore electrical engineering students at the University of New Hampshire the same material that is in this book, which I didn't see until years after getting a Caltech master's degree. So educational times have changed for the better. ...

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