I love it when you go into an exam fully expecting to fail, spend the first ten minutes staring at the exam paper thinking “I don’t even know what equation to start with”, and then getting an epiphany somewhere along the way so you can suddenly derive the answers to everything. Except the one about Gaussian statistics. I don’t really know what those are. Still though, I’m happy.

Statistical mechanics has a lot to do with random fluctuations and entropy. It’s made me realise just how much concepts of order and randomness don’t make complete intuitive sense to me.

Take, for instance, a string of a million “random” numbers. If the set is truly random, that is, each digit has a an equal chance of being any one of the numbers from 0 to 9, then each of those numbers will appear about an equal number of times — one hundred thousand times, in this case. They won’t all be exactly that, but you can go as far as to predict the average number that they’ll differ from that. A child wandering around a house at random should spend an equal ammount of time in each room. (As long as each room is the same size and maybe some other constraints, I suppose.)

Even stranger still, you can tell whether or not a human typed out those numbers or if they were truly randomly generated. A human typing merrily away will at some point think “gee, I haven’t hit 8 in a while”, and will then hit the number 8. But they’ll also think that too many 8′s in a row doesn’t look random enough. Yet, a real random set of numbers will have strings of repeated numbers. There’s a formula to predict how many strings of how many repeats there should be, though I forget what it is. I bet that in a million random numbers, you’ll have at least one part with ten 8′s in a row.

This is how you can tell if the night sky in television or movies is fake or not. If the stars are evenly distributed in the sky, you can bet somebody put those stars in manually. If there are clusters and blank spots, it’s more likely a real starscape or a picture generated by a computer. Or at least made by somebody who knows all this stuff already and wants to make a realistic starscape.

My point is that it doesn’t seem right for it to be possible to predict the properties of a random sequence of numbers. You might get a string of one million 8′s in a row, but the chances of that happening are 1 in ten to the millionth power. I guess that’s the rub. Yet I’m still troubled. How can something really be random if we know what to expect?

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