Sunday, July 15, 2018

Of Hot Hands and Roulette Wheels and Coin Flips: The Fallacy That Became A Fallacy

From Koenfucius, July 6:

Even specialists can fall prey to cognitive errors

I have been tossing a fair coin, and it has come up heads six times in a row. The chance of either heads or tails is 1 in 2, but for this to happen six consecutive times is 1 in 2 to the power 6, or 1 in 64. What is the probability that the next toss will turn up a seventh consecutive head?

The correct answer is of course 1 in 2. Coins don’t have a memory. What happened before cannot influence the current result. We all know that – yet our intuition sometimes leads us to believe differently.

On 18 August 1913, the ball at one of the roulette tables in the Monte Carlo casino had been ending up in a black slot for nearly 20 times in a row. Several gamblers started taking an interest and started putting money on red: after such a long streak of black, red was surely due to come up. And still the ball kept falling on black. People put more and more money on red at each successive turn of the wheel – and kept on losing it, as black kept on coming up. Let’s face it, where would you put your money if you happened to be there, having seen black come up 25 times in a row? Eventually, after an unbroken series of 26 blacks (likelihood: 1 in more than 136 million) the ball finally landed on red.

More recently, it was number 53 in the Italian lotto that had failed to come up for nearly two years and which, apparently, not only drove people bankrupt, but some to their death. The phenomenon has been described long before, though, for example by the early 19th century mathematician Pierre-Simon de Laplace, as Joshua Miller and Andrew Gelman discuss in a fascinating paper.

It is gamblers who give this cognitive illusion – that if something happens less frequently than normal for a period, it will occur more frequently in the future (or vice versa) – its name: the gambler’s fallacy.

Let’s not assume that only compulsive gamblers fall for it, however. If your neighbour is pregnant again, having had five girls, what is the chance of the sixth child being a boy? Did you not, for a moment, feel that it ought to be more than 1 in 2? Given the dry and sunny weather we’ve been experiencing in the UK for weeks now, do you think a dry second half of August is more or less likely than normal?

We treat streaks as if they predict the next outcome. Our human brain is ill-equipped to handle the concept of randomness: we easily see patterns where there are none. The figure below[1] shows the outcome of three sequences of 50 roulette wheel spins (ignoring any zeroes). We know that red and black are equally likely, and we intuitively interpret this as “50% of the outcomes should be black”.  When we see long streaks of one colour, we think that is extraordinary. The universe is out of balance, and a reversal is overdue. Our intuition tells us that after the five blacks in A, a red outcome must be more likely to redress the imbalance. A sequence without long streaks (like C) looks more ‘normal’, but in reality, A and B are the result of genuine spins, while C has been manipulated to generate a reversal 3 out of every 4 times.

The hot hand

There is another phenomenon that is often compared with the gambler’s fallacy. Known as the hot hand fallacy, it was first described in 1985 by Thomas Gilovich, Amos Tversky and Robert Vallone. The ‘hot hand’ refers to a presumed temporary state of a basketball player in which they are more likely to perform better than average, i.e. they will be producing streaks of successful shots. Fans, coaches and players alike widely believe that players are more likely to make a shot after having made the last two or three shots, than after having missed them.

In several studies the researchers failed to find any significant correlation between shots (except for one player). They concluded that the belief in the hot hand in basketball is a cognitive illusion, resulting from the “expectation that random sequences should be far more balanced than they are, and the erroneous perception of a positive correlation between successive shots.” The chance of scoring does not depend on what went on before.

For about 30 years, this conclusion remained largely unquestioned. The hot hand fallacy stood as one of the more robust cognitive errors, even as new research using larger datasets found patterns actually consistent with the hot hand. Issues of measurement and control prevented these studies from determining the magnitude of the hot hand, and so they did not topple the prevailing wisdom.
But in 2016 Joshua Miller and Adam Sanjurjo discovered a fundamental flaw in the reasoning by Gilovich et al. One of their original studies positioned players on various spots on an arc, at a distance from which their shooting percentage was approximately 50%. Each player had to take 100 shots and they found no marked difference in the goal percentages after 1, 2 or 3 hits, or after 1, 2 or 3 misses. Intuitively, this is exactly what you would expect to find if you were to replace the outcomes of the players’ shots by a sequence of 100 coin flips. As successive flips are independent, we expect the percentage of heads that follow a streak of heads to be identical to the percentage of tails that follow a streak of heads.

This is incorrect.
Bafflingly so, but yes, it truly is incorrect. Meet Jack, a hypothetical character from Miller and Sanjurjo’s paper. Jack tosses a coin 100 times. Every time it produces heads, next time he flips he writes down the result. He – like Gilovich and co, and most of us – expects to see approximately 50% of heads. But it is less.

To illustrate why, let’s look at what happens when he tosses the coin just three times. There are eight possible outcomes:...MUCH MORE