Seeking Order in Randomness

Not surprisingly, people interpreted the recent statistical tie in the Florida election in a variety of ways.

Ties seem to bring out our differing mind-sets and preconceptions. I had a visceral illustration of this when Florida Supreme Court Chief Justice Charles T. Wells quoted me in his dissent from last month’s court decision to allow a recount to continue.

The remark, “The margin of error in this election is far greater than the margin of victory, no matter who wins,” in my opinion, lent greater moral weight both to Al Gore’s popular vote victory and to the plurality of Floridians who, I believe, intended to vote for him. It also accorded with the demand for a manual recount.

But have no fear; this piece is not a rehash of the post-election drama. Rather, it investigates the more general question of how people can look at nebulous, inconclusive data and draw very different conclusions from it. One illustration of this is a party game (discussed by philosopher Daniel Dennett in a different context).

Dream Weaving Game

Imagine a group of people at a New Year’s Eve party who choose one person and ask him (or her) to leave the room. The victim is informed that while he is out of the room one of the other partygoers will relate a recent dream to the group. He is also told that on his return to the party, he must try, asking Yes or No questions only, to do two things: reconstruct the dream and figure out whose it was.

The punch line is that no one relates any dream. The partygoers decide to respond either Yes or No to the victim’s questions according to chance or, perhaps, according to some arbitrary rule. Any rule will do and may be supplemented by a non-contradiction requirement so that no answer directly contradicts an earlier one.

The surprising result is that the victim, impelled by his own obsessions, often constructs an outlandish and obscene dream in response to the random answers he elicits. He may also think he knows whose dream it was, but then the trick is revealed to him. (It’s a good way to lose a friend.)

Technically the dream has no author, but in a sense, of course, the victim himself is. His preoccupations dictate his questions, which even if answered negatively at first, frequently receive a positive response when slightly reformulated. These positive responses are then pursued.

Finding Our Own Patterns

A similar explanation may help clarify why silly I Ching sayings or vague horoscopes seem to many to be so apt. Their aptness is self-provided.

They are sufficiently ambiguous to somehow “answer” questions that are implicitly asked of them, and the devotee fabricates these answers into something seemingly appropriate.

The fact is that random events can frequently seem to contain significant patterns. The diagram below is a printout of a random sequence of 250 simulated coin flips, the H’s or T’s each appearing with probability 1/2. Note the number of runs (strings of consecutive H’s or T’s) and the way there seem to be clusters and other patterns.

If you felt you had to account for these patterns, you would have to invent convoluted explanations that would of necessity be quite false.


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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