Both the rules of probability and the law of large numbers describe the regular behavior of chance phenomena in the long run. Psychologists have discovered that the popular understanding of randomness is quite different from the true laws of chance. Most people believe in an incorrect �law of small numbers.� That is, we expect even short sequences of random events to show the kind of average behavior that in fact appears only in the long run.
Try this experiment: Write down a sequence of heads and tails that you think imitates 10 tosses of a balanced coin. How long was the longest string (called a run) of consecutive heads or consecutive tails in your tosses? Most people will write a sequence with no runs of more than two consecutive heads or tails. Longer runs don�t seem �random� to us. In fact, the probability of a run of three or more consecutive heads or tails in 10 tosses is greater than 0.8, and the probability of a run of three or more heads and a run of three or more tails is almost 0.2. This and other probability calculations suggest that a short sequence of coin tosses will often not seem random to most people. The runs of consecutive heads or consecutive tails that appear in real coin tossing (and that are predicted by the mathematics of probability) surprise us. Because we don�t expect to see long runs, we may conclude that the coin tosses are not independent or that some influence is disturbing the random behavior of the coin.
Belief in the law of small numbers influences behavior. If a basketball player makes several consecutive shots, both the fans and his teammates believe that he has a �hot hand� and is more likely to make the next shot. This is doubtful. Careful study suggests that runs of baskets made or missed are no more frequent in basketball than would be expected if each shot were independent of the player�s previous shots. Players perform consistently, not in streaks. (Of course, some players make a higher percent of their shots in the long run than others.) Our perception of hot or cold streaks simply shows that we don�t perceive random behavior very well.
Gamblers often follow the hot-hand theory, betting that a run will continue. At other times, however, they draw the opposite conclusion when confronted with a run of outcomes. If a coin gives 10 straight heads, some gamblers feel that it must now produce some extra tails to get back to the average of half heads and half tails. Not so. If the next 10,000 tosses give about 50% tails, those 10 straight heads will be swamped by the later thousands of heads and tails. No compensation is needed to get back to the average in the long run. Remember that it is in the long run that the regularity described by probability and the law of large numbers takes over. Our inability to accurately distinguish random behavior from systematic influences points out once more the need for statistical inference to supplement exploratory analysis of data. Probability calculations can help verify that what we see in the data is more than a random pattern.
Adapted from:
Moore, David S., McCabe, George P., Duckworth II, William M., and Sclove, Stanley L., "The Practice of Business Statistics - Using Data for Decisions", W.H.Freeman Publishers, New York, 2001, pp.283-284.