This chapter began with a table showing the contrast between what people think random events are like and what they are really like. Here we will explore some of the reasons that people have these erroneous beliefs. It has been well documented that most people—even those who understand that any result of a series of tosses of a fair coin is a random sequence—make errors in their judgements about random sequences. The following is a list of some possible explanations for this tendency. The focus here is not on superstitious beliefs, but on cognitive processes and experiences that might lead a person to hold faulty beliefs. For a more complete examination of erroneous beliefs in gambling, see Wagenaar (1988); Ladouceur and Walker (1996); Kahneman and Tversky (1982); Toneatto (1999); and Toneatto, Blitz-Miller, Calderwood, Dragonetti, and Tsanos (1997).
People will often judge the coin-tossing sequence of H, H, H, H, H, H as being less random than H, T, H, H, T, H, even though the probability of obtaining each of these given sequences is identical: 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 0.015625. Note that this is the probability of getting a specific sequence compared to the probability of getting a second specific sequence. Kahneman and Tversky (1982) call this tendency the representativeness heuristic. People who make this error are often computing the probability of getting 6 consecutive heads compared to every other possible sequence. Most random sequences of heads and tails do not have an easily recognizable pattern. This tends to reinforce the belief that a sequence of all heads is less likely. However, any one specific arbitrary combination of heads and tails has exactly the same chance of occurring as any other specific combination. Another factor that contributes to this error is that there is only one possible way of getting 6 heads and only one possible way of getting 6 tails, while there are a total of 64 possible ways of tossing six coins, 20 of which produce exactly 50% heads (e.g., H, T, T, T, H, H or H, T, H, H, T, T). This gives the illusion that combinations that “look random,” are more likely, but in fact each one of those specific combinations (e.g., H, T, T, T, H, H) has the same chance of occurring as H, H, H, H, H, H.
Another reason for errors in our understanding of randomness may be confusion between the way the word random is used in everyday speech and the way it is used in statistics and mathematics. According to the Merriam-Webster Online Dictionary, the most common meaning of the adjective random is “lacking a definite plan, purpose, or pattern.” It also lists “haphazard” as a synonym (www.m-w.com). Judging solely by its appearance, a sequence of 6 heads in a row might appear to have a pattern. Probability theory, however, is concerned with how the events in a sequence are produced, not in how they appear after the fact.
A third reason is the tendency of the human brain toward “selective reporting”—the habit of seizing on certain events as significant, while ignoring the other neighbouring events that would give the chosen events context and help to evaluate how likely or unlikely the perceived pattern really is. Big or salient events will be recalled better. We recall plane crashes because they are highly publicized. Uneventful flights are ignored. Because of the occasional well-publicized plane crash many people are afraid to fly, even though plane crashes are much rarer than car crashes. Kahneman and Tversky (1982) call this tendency the availability heuristic.
A closely related tendency is for people to underestimate the likelihood of repeated numbers, sequences, or rare events occurring by pure chance. The basic problem is that we do not take into account the number of opportunities for something to occur, so we are often surprised when random chance produces coincidences. As an example, in a class of 35 students, we assume that the chance of 2 people sharing a birthday is very small, say 1 in 365, or maybe 35 in 365 (Arnold, 1978). The actual probability that at least two people will share the same birthday is close to 100% because there are actually (35 x 34) / 2 = 595 possible combinations of people in the class. Because the possible combinations of people (595) exceed the number of days in the year (365), the chance that at least 1 pair of people will share a birthday is surprisingly high.
Our minds are predisposed to find patterns, not to discount them. It is argued that we have evolved the ability to detect patterns because to do so was often essential for survival. For example, if a person was walking in the jungle and saw a pattern of light and dark stripes in the shadows, it would be prudent to assume that the pattern was a tiger and act accordingly. The consequences of incorrectly assuming that the pattern is not a tiger far outweigh those of incorrectly assuming that it is. But when applied to random events, this survival “skill” leads to errors.
Some errors might be the result of the way in which statistics are disseminated. Academics, journalists, advertisers and others often report statistics using terms such as “1 out of every 10,” or “1 death every 25 seconds.” These statements might lead to the impression that the events reported occur in a regular manner.
We learn through experience and logically induce general rules on that basis. If our experience is limited, we may induce the wrong rule. A chance occurrence may lead to false expectations. As a result, a win the first time one plays a game, or a win after some extraneous event, may lead to the formulation of an erroneous general rule. For example, a bingo player reported that she was once about to buy her bingo booklet, but was called away for some reason. Later, she bought her booklet and then won. Now she has a ritual of going back to the end of the line if she does not feel that the serial numbers are lucky, and she reports that this system has worked for her on at least one other occasion.
Natural human reasoning tends to assume that a premise is reversible. That is, given the premises that all As are Bs and all Bs are Cs, the correct conclusion is that all As are Cs. However, people tend to assume that all Cs must also be As. In fact, this is incorrect. All that we can be certain of is that some Cs are As, but there may be many Cs that are not As. This “conversion error” is common and it creates all manner of problems (Johnson-Laird, 1983). Even highly educated individuals frequently make conversion errors. The basic flaw in the law of averages is the error of converting the correct premise “The number of heads and tails even out in the long run” to the incorrect conclusion “Since the number of heads and tails even out in the long run, I should win if I bet on tails.”
Individuals who gamble often think that random events are self-correcting. One possible reason for this is that their experience seems to be consistent with this belief. Closely related to the law of large numbers is the phenomenon of “regression to the mean,” which predicts that exceptional outcomes (e.g., very high or very low scores) will most likely be followed by scores that are closer to the mean. For example, a father who is very tall is more likely to have a son who is shorter than he is, not taller. It is true that a tall man is more likely to have a tall son than a short one, because height is partly under genetic control. However, the random factors that influence height (the recombination of the parents’ genes, nutrition, accidents, diseases, etc.) will tend to pull the son’s height down closer to the average for the general population. The fact is that, by pure chance, there is more room to move down, closer to the mean, than up, away from the mean.
To turn to a gambling example, suppose a coin is tossed 100 times, and 80% are heads. If the coin is tossed another 100 times, the net outcome is more likely to move closer 50% heads than to stay at 80% heads or to increase to 90%. But it is important to understand that regression to the mean does not have to occur: the son could be taller or the next 100 flips could all be heads. But it is more likely that the son’s height or the number of heads and tails will be closer to the mean because the mean is the single most likely outcome. In the context of gambling, regression to the mean might produce the illusion that the random events are “evening out.” Unusual events (long losing streaks or winning streaks) seem to be corrected over time, but in fact they are not corrected, only diluted. The average converges towards the mean; it is not pushed there. But the experience from event to event gives the illusion that it is pushed there by some sort of force.
Increased bets may also play a role in convincing those who gamble that random events are self-correcting because “chasing” works. Doubling a bet after a loss has the interesting effect of increasing the player’s chance of walking away a winner. The rationale behind this practice is again the law of averages. Since people expect random events to correct themselves, doubling after a loss may seem like a good investment strategy. Incremental betting strategies appear to push around random events so they do not look “random” (Turner & Horbay, 2003). Turner (1998) has shown that a doubling strategy would be successful if random events were self-correcting. The chapter “Games and Systems” discusses this betting system and its flaws in more detail. It is enough to note here that most of the time doubling appears to work, thus reinforcing the idea that random events correct themselves. This system usually produces a very slow accumulation of money. Eventually, however, the player experiences a disastrous losing streak
One final reason for errors in judging random events is that our minds tend to segment events in ways that are consistent with what we expect. Given a heads and tails sequence of H, H, H, H, T, H, H, H, T, T, T, T, H, we are likely to divide this string into a segment in which H was more likely to appear (H, H, H, H, T, H, H, H) and one in which T was more likely to appear (T, T, T, T, H). This segmentation process is very often used by sports commentators (e.g., “The Blue Jays have now won 5 of their last 6 games,” or “A player has struck out 11 times in his last 15 at bat”). In segmenting the sequence this way, it is very easy to convince oneself that tails did in fact come up more often, to correct for the excess of heads. As noted above, our minds are predisposed to find patterns, not to discount them.