Law of Small Numbers
Having learned the probabilities associated with certain sequences of coin tosses, I offer some particularly important questions about the likelihood of certain clustered outcomes.
Caution: Even if you believe you have answered enough coin-tossing questions, do not skip these questions.
Question 6.1.1 Jack and Jill have played a coin-tossing game each day for 1,000 consecutive days spanning most of the past three years. Jack always bet heads; Jill always bet tails. It was a fair coin and Jack and Jill were equally likely to win.
Jack was ahead on any given day if the tally of the number of heads exceeded the number of tails. Jill was ahead on any given day if the tally of the number of tails exceeded the number of heads.
Which of the following is the most likely description of their game?
a. Over time the lead changed frequently between Jack and Jill as their winning percentages seesawed back and forth between 48 and 52 percent.
b. One of the players moved ahead quickly—and stayed ahead— for more than 96 percent of the tosses.
As discussed earlier, on any toss of a fair coin the likelihood of the coin's landing heads versus tails is exactly 50-50. Unequivocally, the more tosses the more the percent deviation from the expectation shrinks.
Yet even in a perfectly random game such as coin tossing winners and losers emerge. Moreover, once the winners are ahead they are unlikely to relinquish their winning positions. Talk about counterintuitive! The correct answer to Question 6.1 is "b"—one of the players moved ahead quickly—and stayed ahead—for more than 96 percent of the tosses. The lesson to be learned from this example is that even though one player appears to have superior skill it is an illusion. You have been fooled into thinking there is a pattern in a sequence of undeniably chance outcomes.
Question 6.2.2 You and a friend toss a coin once a day. You always bet on heads; your friend always bets on tails. After about two months you and your friend both have a better-than-even chance of winning how many tosses in a row?
Question 6.3. Anyone who watches basketball knows that players have "hot" and "cold" streaks. That is, after making a couple of shots, hot players get in a groove and are more likely to score successive points. Conversely, players who miss several shots are said to be "cold" and are less likely to score on successive shots.
The correct answer to Question 6.2 is "e"—after 60 coin-tossing games, two players each have a better-than-even chance of winning five tosses in a row.
Turning to the basketball question, a study by Gilovich, Vallone, and Tversky found that 91 percent of the knowledgeable basketball fans who were interviewed thought that a player has a "better chance of making a shot after having just made the previous two or three shots than after having just missed the previous two or three shots."3 (Given that the fans who were interviewed were from Boston and Philadelphia, it is easy to understand that the word "fan" is derived from the word "fanatic.")
Contrary to what these fans believe—and the players whose shooting records were analyzed believe—the data show that basketball players are no more likely to make a shot after making their last one, two, or three shots than after missing their last one, two, or three shots. Perhaps hot streaks are perfectly compensated for by the tendency of "hot" players to attempt more difficult shots or to be more closely guarded.
Even though the relevant statistical tests indicate that there are no hot or cold streaks, fans simply refuse to believe the analysis. Siding with the fans, the analysis did not take into account the difficulty of the shot and the amount of defensive pressure.
One segment of basketball where shot difficulty and the amount of defensive pressure are not factors (particularly if you segregate home and away games) is free throws. Gilovich, Vallone, and Tver-sky's analysis of two seasons of free-throw statistics by the Boston Celtics showed that players did not run hot or cold. On average, Boston Celtics players made 75 percent of their second free throws after making their first and 75 percent after missing their first.
Hence, the correct answer to Question 6.3 is "b"—false. There is no evidence that basketball players have hot or cold streaks when shooting free throws.
The lesson here is that we expect random sequences—such as those produced by coin tosses—to alternate between heads and tails; however, in truth, truly random sequences have far more repetitions of one outcome than our intuition leads us to expect. Streaks of four, five, or six heads or tails in a row clash with our expectations for alternating heads then tails then heads sequences. Yet in a series of only 20 coin tosses there is a 50-50 chance of getting four heads in a row, a 25 percent chance of five in a row, and a 10 percent chance of a streak of six.4
I should emphasize that this research in no way implies that whether a basketball shot goes in the hoop is a random phenomenon. Whether a high or low percentage of a player's shots go in the hoop is a function of the player's offensive skill and the defensive skill of the players on the other team.
What the research here does show is that whether a basketball shot goes in the hoop or does not go in the hoop cannot be derived from studying the in-the-hoop or the not-in-the-hoop sequences in the player's previous shots.
Question 6.4.5 I show you and a large group of diehard basketball fans the following sequence:
OXXXOXXXOXXOOOXOOXXOO
You and the fans are told the Xs stand for shots made and the Os stand for the shots missed in an actual National Basketball Association game.
What percentage of the fans believe there are hot streaks in the data?
Do you believe there are hot streaks in the sequence?
Most basketball fans see hot streaks in the sequence. In Gilovich, Vallone, and Tversky's study using this sequence 61 percent of the fans saw hot streaks. Frankly, I would have guessed a much higher percentage. After all, in the first eight symbols there are six Xs and only two Os. If Alan Iverson of the Philadelphia 76ers made six of his first eight shots, the announcer would certainly say something like "Alan is on fire in the opening minutes." The rub is, as you may have guessed, that the Gilovich, Vallone, and Tversky sequence was constructed to meet the precise definition of random.
Behavioral economists call our tendency to see patterns where none exist a "clustering illusion." Thomas Gilovich emphasizes that the importance of this insight "lies in the inescapable conclusion that our difficulty in accurately recognizing random arrangements of events can lead us to believe things that are not true and to believe something is systematic, ordered, and 'real' when it is really random, chaotic, and illusory."6
Again quoting Gilovich, "We are predisposed to see order, pattern, and meaning in the world; we find randomness, chaos, and meaninglessness unsatisfying. Human nature abhors a lack of predictability and the absence of meaning."7
Question 6.5.8 One of the following sequences is an actual sequence that was derived from spinning the needle on the (unbiased) wheel shown in Figure 6.1. (R stands for red and G stands for green.) The two other sequences are made up. Notice that the likelihood the needle will stop on green is four out of six (66.7 percent); the likelihood the needle will stop on red is two out of six (33.3 percent).
Which of the following series has the highest probability of being the real sequence?
b. GRGRRR.
c. GRRRRR.
Question 6.6. Note that the RGRRR sequence in choice "a" is imbedded in the G R G R R R sequence in choice "b." Does this change your answer?
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