What we mean by random

0:00Here is the shape of every situation this subject is built for. You know two things, and you cannot know a third. You know the whole list of ways it could turn out. You know that one of them is going to happen. What you do not know is which. That is not ignorance about the situation - the list is in front of you, complete. It is ignorance about this one occasion.

0:26And the word for a set-up like that is random, which means something far narrower, and far more useful, than the way the word usually gets thrown about. Two examples, chosen because their lists are short enough to write out in full. A coin: heads, tails. Two items, and that is the whole list. A die: one, two, three, four, five, six. Six items, and again that is all of them.

0:56Now notice what a list like that rules out. A die will not come up seven. Seven is not on the list, so its chance is zero - not small, zero. Random does not mean anything at all could happen; it means exactly one of these will, and you cannot say which. The list is a promise as much as it is a menu. So here is what randomness asks for, and it comes in three parts.

1:26One: you can run the set-up again. Two: when you do, the result need not come out the same. Three: nobody can name the result before it happens. All three together, and that matters. It is easy to take in only the third, because the not-knowing feels like the whole idea. It is not. The first part is the one people walk straight past, and it is the one that makes any mathematics possible at all - something that happens once and can never be repeated is not what this is built to handle.

2:05Are all three really needed, or is that just being thorough? There is a way to find out. Take the three parts and consider every way they could hold or fail - eight ways in all. Exactly one of the eight is a random experiment: the one where all three hold. Now strike out a part, and see what the weakened version lets in. Strike the first, and a draw held once and never held again walks straight through.

2:35Strike the second, and so does a lamp that lights every single time. Strike the third, and in comes a machine that goes heads, tails, heads, tails. Each part keeps out exactly one thing, and no part is doing another's job. That last one is worth a longer look, because it is a trap. The machine alternates - heads, tails, heads, tails - for as long as you care to watch.

3:04Run it twenty times. You get ten heads and ten tails, which is exactly the count you would expect from a fair coin over twenty tosses. So counting the results cannot tell the machine from the coin. But anyone who has spotted the pattern calls all twenty correctly, where a guesser on a real coin averages ten. The machine varies, and it repeats, and it is completely predictable. Two parts out of three is not enough.

3:37Back to the draw for a moment. Every student's name on a slip, one slip each, all folded the same way, mixed, and one pulled out without looking. In a school of five hundred, each chance is one in five hundred, and every one of those chances is the same. Now ask where that equality came from. Not from the students - nothing about them entered the argument anywhere. It came from the folding.

4:06Bend one slip so it is three times as easy to catch, and that slip's chance becomes three in five hundred and two while everybody else drops to one in five hundred and two. Equal chances are earned by the set-up, and they are lost with it. Now the hard case: rain. The atmosphere is not mysterious. It runs on ordinary physics, and what it does next turns on four things you could in principle go out and measure - temperature, humidity, wind and pressure.

4:42None of that is hidden. And yet tomorrow's weather is treated as random, which sounds like a contradiction until you look at what randomness actually asked for. It asked that nobody can name the result beforehand. It never asked that nothing caused it. Those are two different claims, and the difference shows up on something far simpler than a sky. Take a coin spun in the air, and suppose the face it shows is settled entirely by how long it spins - no mystery, just a rule.

5:20Spin it for half a second at five half-turns a second and it has completed two half-turns, so it comes up the way it started. Entirely determined. Now try predicting it. At twelve half-turns a second, a stopwatch good to a thousandth of a second pins the face down ninety-seven times in a hundred. Good to a hundredth, seventy-five. Good to a twentieth, never once. And the same stopwatch that reads a slow spin loses a fast one: ninety-five in a hundred at two half-turns a second, twenty at forty.

6:01So the rule is perfectly determined and, past a certain speed, not predictable with anything you can actually measure with. Here is how badly that scales. Take a system that pulls small differences apart, start it twice from points differing in the twelfth decimal place, and after thirty-seven steps the two runs are nowhere near each other. Measure ten thousand times more finely and you buy fourteen more steps. Ten thousand times better again buys the same fourteen.

6:37Precision is not the thing that was missing. So what is left, when the result is not? The likelihood. That is the quantity that survives, and it is what everything after this is about. Which is also why being right once is worth nothing. Call heads before a toss and you will be right about half the time. Call it four times and you will be right at least once in ninety-three cases out of a hundred.

7:07Nearly unavoidable - and it tells you nothing about the coin, and nothing about the caller. Anything that easy to achieve cannot be evidence of anything. One confusion to clear before moving on, because everything so far has quietly encouraged it. Coins, dice, slips in a box: all random, and all with equally likely outcomes. It is very easy to come away thinking random means fifty-fifty, or one in six.

7:41It does not. Think of a driver turning a key - the car starts, or it does not. Two outcomes, a known list, an unknown result: random by every part of the definition. And nobody believes those two are equally likely. Random is one property, equally likely is another, and travelling together often does not make them the same thing. Which brings us to why a coin decides who bats first, and there are three conditions behind it.

8:15The advantage is real, so it has to go to somebody. The two outcomes are equally likely, so neither side is favoured before the coin leaves the hand. And neither captain can influence it or see it coming, which is why it happens in the open with both sides watching. Take away the second and you have a weighted coin; take away the third and you have a fair coin tossed out of sight.

8:44Either way the method stops being fair, so equal chances alone were never enough on their own. Notice too that the test never asks how the coin landed - the captain who loses has not been treated unfairly. And last, the quiet thing all this buys: you can run it again. Twenty tosses can leave any of a million different records, and the chance that two full runs leave the same one is about one in a million.

9:15That is not a nuisance to be tidied away - repeating is how a likelihood gets measured, and measuring one is what comes next.