John Paulos - INNUMERACY - Mathematical Illiteracy and Its Consequences

The simple answer follows, so don't read on if you want to think a bit. The man's more frequent trips to the Bronx are a result of the way the trains are scheduled. Even though they each come every twenty minutes, the schedule may be something like the following: Bronx train, 7:00; Brooklyn train, 7:05; Bronx train, 7:20; Brooklyn train, 7:25; and so on. The gap between the last Brooklyn train and the next Bronx train is fifteen minutes, three times as long as the five-minute gap between the last Bronx train and the next Brooklyn train, and thus accounts for his showing up for three-fourths of his dates in the Bronx and only for one-fourth of his Brooklyn dates.

Countless similar oddities result from our conventional ways of measuring, reporting, and comparing periodic quantities, whether they be the monthly cash flow of a government or the regular daily fluctuations in body temperature.

Imagine flipping a coin many times in succession and obtaining some sequence of heads and tails; say, HHTHTTHHTHTTTHTTHHHTHTTHHTHHTTHTHHTTHHTHTHHHHTHHHTT. If the coin is fair, there are a number of extremely odd facts about such sequences. For example, if one were to keep track of the proportion of the time that the number of heads exceeded the number of tails, one might be surprised that it is rarely close to half.

Imagine two players, Peter and Paul, who flip a coin at the rate of once a day and who bet on heads and tails respectively. Peter is ahead at any given time if there've been more heads up until then, while Paul is ahead if there've been more tails. Peter and Paul are each equally likely to be ahead at any given time, but whoever is ahead will probably have been ahead almost the whole time. If there have been one thousand coin flips, then if Peter is ahead at the end, the chances are considerably greater that he's been ahead more than 90 percent of the time, say, than that he's been ahead between 45 percent and 55 percent of the time! Likewise, if Paul is ahead at the end, it's considerably more likely that he's been ahead more than 96 percent of the time than that he's been ahead between 48 percent and 52 percent of the time.

Perhaps the reason this result is so counterintuitive is that most people tend to think of deviations from the mean as being somehow bound by a rubber band: the greater the deviation, the greater the restoring force toward the mean. The so-called gambler's fallacy is the mistaken belief that because a coin has come up heads several times in a row, it's more likely to come up tails on its next flip (similar notions hold for roulette wheels and dice).

The coin, however, doesn't know anything about any mean or rubber band, and if it's landed heads 519 times and tails 481 times, the difference between its heads total and its tails total is just as likely to grow as to shrink. This is true despite the fact that the proportion of heads does approach 1/ 2as the number of coin flips increases. (The gambler's fallacy should be distinguished from another phenomenon, regression to the mean, which is valid. If the coin is flipped a thousand more times, it is more likely than not that the number of heads on the second thousand flips would be smaller than 519.)

In terms of ratios, coins behave nicely: the ratio of heads to tails gets closer to 1 as the number of flips grows. In terms of absolute numbers, coins behave badly: the difference between the number of heads and the number of tails tends to get bigger as we continue to flip the coin, and the changes in lead from head to tail or vice versa tend to become increasingly rare.

If even fair coins behave so badly in an absolute sense, it's not surprising that some people come to be known as "losers" and others as "winners" though there is no real difference between them other than luck. Unfortunately perhaps, people are more sensitive to absolute differences between people than they are to rough equalities between them. If Peter and Paul have won, respectively, 519 and 481 trials, Peter will likely be termed a winner and Paul a loser. Winners (and losers) are often, I would guess, just people who get stuck on the right (or wrong) side of even. In the case of coins, it can take a long, long time for the lead to switch, longer often than the average life.

The surprising number of consecutive runs of heads or tails of various lengths give rise to further counter-intuitive notions. If Peter and Paul flip a fair coin every day to determine who pays for lunch, then it's more likely than not that at some time within about nine weeks Peter will have won five lunches in a row, as will have Paul. And at some period within about five to six years it's likely that each will have won ten lunches in a row.

Most people don't realize that random events generally can seem quite ordered. The following is a computer printout of a random sequence of Xs and Os, each with probability 1/ 2 .

oxxxoooxxxoxxxoxxxxo

oxxoxxoxooxoxooooxox

xoooxxxoxoxxxxxxxxxo

xxxoxoxxxxoxooxxxooo

xxxxxooxxoooxxooooox

xooxxxxxxoxxxxooxxxx

ooxxoxxooxxoxoxooxxx

oxxoxxxxoxxoxxxxxxxx

xoxxxxxoooooxooxxxoo

xxxxooxooxoxxxoxxxxo

oooxoxoxxoxxxooxxooo

oxxxxxooooxxxxoxxoox

xxxxxoxxoooooooxoxxx

xxoooxxoxxxooooxoxox

ooxxxxoxoxxxoxxooxxo

xooxooxxxoxx

Note the number of runs and the way there seem to be clumps and patterns. If we felt compelled to account for these, we would have to invent explanations that would of necessity be false. Studies have been done, in fact, in which experts in a given field have analyzed such random phenomena and come up with cogent "explanations" for the patterns.

With this in mind, think of some of the pronouncements of stock analysts. The daily ups and downs of a particular stock, or of the stock market in general, certainly aren't completely random as the Xs and Os above are, but it's safe to say that there is a very large element of chance involved. You might never guess this, however, from the neat post hoc analyses that follow each market's close. Commentators always have a familiar cast of characters to which they can point to explain any rally or any decline. There's always profit-taking or the federal deficit or something or other to account for a bearish turn, and improved corporate earnings or interest rates or whatever to account for a bullish one. Almost never does a commentator say that the market's activity for the day or even the week was largely a result of random fluctuations.

The clumps, runs, and patterns that random sequences evince can to an extent be predicted. Sequences of heads and tails of a given length, say twenty flips, generally have a certain number of consecutive runs of heads. A sequence of twenty coin flips which resulted in ten heads followed by ten tails (HHHHHHHHHHTTTTTTTTTT) is said to have just one run of heads. A sequence of twenty coin flips which resulted in heads and tails alternating (HTHTHTHTHTHTHTHTHTHT) is said to have ten runs of heads. Both these sequences are unlikely to be randomly generated. A sequence of twenty flips with six runs of heads (say, HHTHHTHTTHHHTTHHTTHT) is more likely to have been generated at random.

Criteria like this can be used to determine how likely it is that sequences of heads and tails or Xs and Os or hits and misses are randomly generated. In fact, psychologists Amos Tversky and Daniel Kahneman have analyzed the sequences of hits and misses of professional basketball players whose basket-making was about 50 percent and found that they seemed to be completely random -that a "hot hand" in basketball, one that would result in an inordinate number of long streaks (runs) of consecutive baskets, just didn't seem to exist. The streaks that did occur were most likely due to chance. If a player attempts twenty shots per night, for example, the probability is surprisingly almost 50 percent that he will hit at least four straight baskets sometime during the game. There's a 20 percent to 25 percent probability that he will achieve a streak of at least five straight baskets sometime during the game, and approximately a 10 percent chance that he will have a streak of six or more consecutive baskets.