Seeking Order in Randomness

An easy exercise further illustrates this. Take a piece of white paper and partition it into small squares so that it looks like a checkerboard. Flip a coin and color the upper left corner red if it lands heads and blue if it lands tails. Proceed to the next square and flip and color again. When you’ve colored the whole paper, look it over and note the patterns and connected clumps of similarly colored squares.

The Illusion of Order

The so-called arc sine law in probability theory provides another way in which random events can give the illusion of order. Consider two people, Henry and Toni, who flip a coin at a steady pace and bet on heads and tails, respectively.

Henry is ahead at any given time if there have been more heads up until that time, and Toni is ahead at any given time if there have been more tails. Henry and Toni are each equally likely to be ahead at any given time, but — and this is the counterintuitive part — one of the contestants will probably be in the lead most of the time. If there are 6 million coin flips, for example, the chances are considerably greater that Henry will be in the lead more than 90 percent of the time, say, than that he will be in the lead between 45 percent and 55 percent of the time.

Likewise, it’s considerably more likely that Toni will be in the lead more than 98 percent of the time than that she will be in the lead between 49 percent and 51 percent of the time. (Other factors are involved, but this would help explain why one candidate might lead during virtually the whole counting process in an election that was nevertheless a statistical tie.)

In conclusion, uncertain information, coincidences, and statistical ties provide fertile ground for all sorts of theories, narratives, and just-so stories. No doubt that 2001 will bring its share of them. In fact, 2001 is 3*667, and 667 is the closest whole number to 666.666 and also equals 23*29, which means that .... Oh, enough of this.

Happy New Year!

Professor of mathematics at Temple University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on appears on the first day of every month.

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