3.4 Some Philosophical Applications

Streaks

In the previous section we looked at the probability of getting a certain number of heads in a number of flips. Look at the following two sequences:

1. HTTHTHHTTHTHTTHHHTHHTTHHTHHTTHTHHTHHTTHTTHHHTHTHTT
2. HHTHHHTTTTTTTHTHTTHTTTHTHTHHTHTTHTTTHHTTTHHHHTHHHH

One of these sequences was generated from actually flipping a coin 50 times. The other one is from a person pretending to flip a coin, and writing down a sequence that they thought would look like a random flipping of a coin. Which one is which? While many people think that sequence 1 looks more “random” (i.e. it seems to flip around a lot), sequence 2 is actually the random sequence.

One of the truly unintuitive things about real random sequences, as opposed to designed sequences, is that there are long runs or streaks. Why is this? The general solution is beyond this book but we can think about it this way. Although a sequence of, say, 5 heads in a row is very unlikely ($$P\left(\text{5 heads in a row}\right)=(1/2{)}^5=0.03$$), there are many opportunities for such a sequence somewhere within a sequence of 50. Because of these many opportunities, this raises the probability from 3% (the probability of 5 heads in a row in 5 flips), to over 55%, the probability of finding 5 heads in a row somewhere in 50 flips. Streaks of 6 heads in a row occur nearly one third of the time in 50 flips, or over half the time if you consider a run to be either heads or tails. Even streaks of 9 heads or tails in a row, in 50 flips, are not extremely unlikely!

Gambler's Fallacy

When we look at a sequence of real coin flips, like:

• HHTHHHTTTTTTT

and we ask about the probability of flipping heads in the next flip, it is common to (mistakenly!) reason that, because we've seen 7 tails in a row, then the next flip is more likely to be heads. However, this is not the case for two reasons:

1. long streaks are common in completely fair and random sequences - so observing a streak of 7 tails does not contribute much to one's confidence that we are looking at a rigged coin or one that has changed its probability properties.
2. the process of flipping a coin is independent each time, nearly by definition, and thus the result of one flip cannot influence the result of the next flip.

The faulty, but intuitive, reasoning goes by the name of the Gambler's Fallacy and appears in many places. We can ask a question:

How could we tell the difference between a random, independent sequence and one where the events are not independent, where the next flip depended on a previous flip?

We'll have to return to this question later, when we consider model comparison, but roughly, one would have to look at all pairs of events to see if one pair (say heads-tails) occurs more frequently (even if only by a little) than another pair (say heads-heads).

In a total fit of irony, casino slot machines do not produce independent winnings - they are programmed so that if you've lost many times, then that machine is a little less likely to lose the next time. In effect, at gambling houses they train the gamblers in the Gambler's Fallacy!

The Hot Hand - Correlations in Random Sequences

Some work by Tversky and Gilovich¹ looks at the following issue in the sport of basketball: there are times when it seems as if basketball players have a “hot hand” - they are on a shooting streak. Tversky and Gilovich looked at how basketball fans perceived streaks, by having them rate sequences of shots as random shooting or streak shooting. Most (65%) of the respondents classified artificially generated, purely random sequences as streak shooting. In real data, they discovered that the actual probability of “making a given shot (i.e. a player's shooting percentage) is unaffected by the player's prior performance.” We examine this effect in a later section (see Example 9.11 on page 172) where we explore the quantitative procedure for assessing this conclusion. It is enough here to note the large difference between the perception of the sequence and the likely cause of the sequence, and thus the need to always be vigilant against faulty perceptions. Tversky and Gilovich insist that “their observations do not tell us anything general about sports, but it does suggest a generalization about people, namely that they tend to 'detect' patterns even where none exist.”

What we have here, again, is the general perception that long sequences are somehow not “random,” when in fact the opposite is the case. People have a natural tendency to see patterns in random data, to infer order where there is none, and to ascribe importance to the appearance of pattern. It is the role of statistical inference in general to provide the tools to properly handle the distinction between random effects and patterns, and to retune our intuitions.

Regression Toward the Mean

There is a peculiar phenomenon referred to as regression toward the mean, which often is misinterpreted and leads to failures of proper statistical inference. It can be seen in a simple example. Imagine that we “test” a number of students by having them guess the results of a coin flip. Clearly this will be entirely luck, because the coin flip has no pattern. If a student guesses the results of 50 flips, there will be an expectation of getting 25 correct. Here we simulate 20 students each “predicting” the result of 50 flips, the results shown in Table 3.1. The test is done twice, and we will look at a particular subset presently. One can, by eye, see that most of the students get around 25 correct - exactly as expected from random performance.

Now, imagine that we look at the top five coin flip predictors on the first round. Will they do better or worse in the the second round? What about the bottom five coin flip predictors? The results of these two cases are summarized in Table 3.2. The pattern, even in this small sample, is quite clear:

1. Those that did the best the first time did worse the second (on average)
2. Those that did the worst the first time did better the second (on average)

One might be tempted (had you not known that this is artificial data, and completely random) to interpret this as a causal pattern, e.g. “the students that did better the first time, grew over-confident the second time,” “the students that did worse the first time, worked harder to improve the second time,” etc... This interpretation of the results by students has been observed in the classroom. However, it runs into serious trouble when the data is something like the heights of children compared to their parents - the tallest parents tend to have children shorter than they are, the shortest parents tend to have children taller than they are, a pattern first quantified by Galton in 1869. He noted that clearly the children are not trying to be tall, so effort is not a good explanation for the pattern.

What is happening here is that, if the process is dominated by luck or simple random variation, then outliers occur, but are rare. Thus a particularly high value will likely be followed by a lower value - closer to the mean. The tendency is to regress toward the mean in processes dominated by luck. This can be confused with the Gambler's Fallacy discussed earlier, where flipping 3 heads in a row doesn't give you any information about flipping another heads - it is not more likely to be tails. Part of the difference is that we are dealing with a process that has many possible values, not just two, and thus we can have a mean value, and outliers.

When each of these ideas is applied to sports, the weather, or business there are some interesting conclusions.

1. even when the process is entirely random, long streaks occur - and are often misinterpreted as an increase in the probability of the event.
2. when a person performs very well at their job (a number of successful business decisions, a high batting average, etc...) they will often do worse the next year - and again many are surprised, and interpret the result as the person “losing their touch” - when in fact, they may just have been lucky for a bit, and are now performing closer to their typical average level.
3. when one has a particularly bad winter, it may be more likely that the next winter won't be quite do bad - due entirely to regression to the mean. It may, however, be part of a larger pattern (e.g. a large-scale climate oscillation, such as El Ni\ no) and the probability of another bad winter might be higher. In order to tell the difference, we need to construct reasonable models of the phenomena, test those models with predictions, and apply those models into the future. At each step, we need to be careful not to jump to the conclusion of the existence of a pattern too quickly.

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.