Which plot is random?

The exhibit shows two seemingly similar patterns – but one of them is random and one isn’t. Can you tell which is which? More in a moment.

I’ve taken these plots from Steven Pinker’s new book, The better angels of our nature: why violence has declined. It’s in a Chapter titled The statistics of deadly quarrels where he discusses the statistical patterning of wars and takes a small detour into “a paradox of utility”, specifically our tendency to see randomness as regularity with little clustering.

This cognitive illusion has relevance to all disciplines but is of particular interest to anyone interested in spatial issues, as a couple of these examples show.

Professor Pinker, who’s a psychologist at Harvard, cites the example of the London blitz, when Londoners noticed a few sections of the city were hit by German V-2 rockets many times, while other parts were not hit at all:

They were convinced that the rockets were targeting particular kinds of neighborhoods. But when statisticians divided a map of London into small squares and counted the bomb strikes, they found that the strikes followed the distribution of a Poisson process—the bombs, in other words, were falling at random. The episode is depicted in Thomas Pynchon’s 1973 novel Gravity’s Rainbow, in which statistician Roger Mexico has correctly predicted the distribution of bomb strikes, though not their exact locations. Mexico has to deny that he is a psychic and fend off desperate demands for advice on where to hide

Another example is The Gambler’s Fallacy – the belief that after a long run of (say) heads, the next toss will be tails:

Tversky and Kahneman showed that people think that genuine sequences of coin flips (like TTHHTHTTTT) are fixed, because they have more long runs of heads or of tails than their intuitions allow, and they think that sequences that were jiggered to avoid long runs (like HTHTTHTHHT) are fair

The exhibit above shows a simulated plot of the stars on the left. On the right it shows the pattern made by glow worms on the ceiling of the famous Waitomo caves, New Zealand. The stars show constellation-like forms but the virtual planetarium produced by the glow worms is relatively uniform.

That’s because glow worms are gluttonous and inclined to eat anything that comes within snatching distance, so they keep their distance from each other and end up relatively evenly spaced i.e. non-randomly. Says Pinker:

The one on the left, with the clumps, strands, voids, and filaments (and perhaps, depending on your obsessions, animals, nudes, or Virgin Marys) is the array that was plotted at random, like stars. The one on the right, which seems to be haphazard, is the array whose positions were nudged apart, like glowworms

Thus random events will occur in clusters, because “it would take a non-random process to space them out. The human mind has great difficulty appreciating this law of probability”.

By the way, this is an interesting book. As summarised by Peter Singer in his review for the New York Times, “the central thesis of “Better Angels” is that our era is less violent, less cruel and more peaceful than any previous period of human existence:

The decline in violence holds for violence in the family, in neighborhoods, between tribes and between states. People living now are less likely to meet a violent death, or to suffer from violence or cruelty at the hands of others, than people living in any previous century.

Even if you don’t buy his big thesis, this is a fascinating journey through the history of civilisation, philosophy, psychology, and much more.

To finish, here’s a puzzle Professor Pinker poses at the start of the chapter (he reckons almost no one gets it right. The answer is in the comments):

Suppose you live in a place that has a constant chance of being struck by lightning at any time throughout the year. Suppose that the strikes are random: every day the chance of a strike is the same, and the rate works out to one strike a month. Your house is hit by lightning today, Monday. What is the most likely day for the next bolt to strike your house?

I’ve looked at interesting statistical “stories” before –e.g. see Can selection bias shoot down an argument.