who recently posted a nice review of this podcast on iTunes. Now,

let's get started.

Suppose you try this experiment: you create a small maze with two

possible paths, only one of which leads to some cheese. You roll some

dice to create a 60% probability that the cheese will be on the left

path, and a 40% probability that the cheese will be on the right. You

run a thousand or so trials with a single mouse, and then watch what

he does for the next hundred trials. How well do you think he would

do? As it turns out, in this experiment, the mouse quickly figures

out that the left path has cheese a lot more often than the right, and

after a sufficient number of trials has occurred, he will always head

left. This means that on average he will get cheese in 60 of the 100

new trials, a pretty good result.

Now print out the history of the mouse's thousand initial trials,

and show it to a human subject, without telling them your formula for

deciding where to place the cheese. Ask the human to recommend what

the mouse should do for the next hundred trials. Experiments have

shown that the human will recommend some pattern that involves both

the left and right paths at varying times, and only score about 52%

here-- a significantly worse result than the mouse gets! What's going

on?

The problem is that we have a natural human tendency to look for

patterns. It's virtually a psychological certainty that if you stare

at a large number of random values, you will start to hypothesize

patterns there. Suppose there is a run of seven right-cheeses

starting at trial 500-- maybe this means that every 500th iteration,

there will be seven right-cheeses in a row. Or maybe you see that of

the 15 perfect even squares below 1000, all had a left-cheese, so that

must mean left cheeses are associated with perfect sequares. While the

chance of any particular pattern like this are tiny, you can be sure

that after the fact, you'll be able to fit some pattern to the random

results. And thus you'll miss the forest, the overall frequency and

the general randomness, for the arbitrary trees that pop up.

In an experiment with rats, this is just amusing. But when you

pick up a newspaper and read about real-life predictions, you should

always keep this experiment in mind. Is some stock guru really

describing a useful investment strategy, or is he just back-fitting

some weird theory on random luck? Was some terrorist attack due to a

foreign policy decision taken last week, or was it just a coincidence?

Will eating shark-cartilage-wrapped raw brussels sprouts floating in a

mixture of red wine and castor oil at every meal protect you from

cancer of the big toe, or were these patterns just back-fit onto a

random study? Not that any of these theories are necessarily wrong--

but you need to think about the effects of true randomness in order to

make an informed decision.

And this has been your math mutation for today.

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