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Chaos, Bounds, and Randomness Help

By — McGraw-Hill Professional
Updated on Apr 25, 2014

Introduction to Chaos, Bounds, and Randomness

Have you ever noticed that events seem to occur in bunches? This is more than your imagination. A few decades ago, this phenomenon was analyzed by Benoit Mandelbrot, an engineer and mathematician who worked for International Business Machines (IBM). Mandelbrot noticed that similar patterns are often found in apparently unrelated phenomena, such as the fluctuations of cotton prices and the distribution of personal incomes. His work gave birth to the science of chaos theory.

Was Andrew "Due"?

In the early summer of 1992, south Florida hadn't had a severe hurricane since Betsy in 1965. The area around Miami gets a minimal hurricane once every 7 or 8 years on the average, and an extreme storm once or twice a century. Was Miami ''due'' for a hurricane in the early 1990s? Was it ''about time'' for a big blow? Some people said so. By now you should know enough about probability to realize that 1992 was no more or less special, in that respect, than any other year. In fact, as the hurricane season began in June of that year, the experts predicted a season of below-normal activity.

The so-called ''law of averages'' (which is the basis for a great deal of misinformation and deception) seemed to get its justice on August 24, 1992. Hurricane Andrew tore across the southern suburbs of Miami and the Everglades like a cosmic weed-whacker, and became the costliest hurricane ever to hit the United States up to that date. Did the severity of Andrew have anything to do with the lack of hurricanes during the previous two and a half decades? No. Did Andrew's passage make a similar event in 1993 or 1994 less likely than it would have been if Andrew had not hit south Florida? No. There could have been another storm like Andrew in 1993, and two more in 1994. Theoretically, there could have been a half dozen more like it later in 1992!

Have you ever heard about a tornado hitting some town, followed three days later by another one in the same region, and four days later by another, and a week later by still another? Have you ever flipped a coin for a few minutes and had it come up ''heads'' 18 times in a row, even though you'd normally have to flip it for days to expect such a thing to happen? Have you witnessed some vivid example of ''event-bunching,'' and wondered if anyone will ever come up with a mathematical theorem that tells us why this sort of thing seems to happen so often?

Slumps and Spurts

Athletes such as competitive swimmers and runners know that improvement characteristically comes in spurts, not smoothly with the passage of time. An example is shown in Fig. 7-7 as a graph of the date (by months during a hypothetical year) versus time (in seconds) for a hypothetical athlete's l00-meter (100-m) freestyle swim. The horizontal scale shows the month, and the vertical scale shows the swimmer's fastest time in that month.

Fig. 7-7. Monthly best times (in seconds) for a swimmer whose specialty is the 100-m freestyle, plotted by month for a hypothetical year.

Note that the swimmer's performance does not improve for a while, and then suddenly it does. In this example, almost all of the improvement occurs during the summer training season. That's not surprising, but another swimmer might exhibit performance that worsens during the same training season. Does this irregularity mean that all the training done during times of flat performance is time wasted? The coach will say no! Why does improvement take place in sudden bursts, and not gradually with time? Sports experts will tell you they don't know. Similar effects are observed in the growth of plants and children, in the performance of corporate sales departments, and in the frequency with which people get sick. This is ''just the way things are.''

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