Probability as a measurement, not a guess

0:00Here are three questions. Will it rain tomorrow? Will our team win on Saturday? Will my name come out of the box at the school draw? You could ask a hundred people and collect a hundred answers, and not one of them can tell you yes or no. That is not because the questions are badly asked. It is because the answers have not happened yet. Mathematics is supposed to be the place where questions have answers.

0:30So what does it do with a question like these? It stops trying to say what will happen, and starts trying to say how much. Look at what those three questions have in common. In every one of them you know the whole list of things that could happen; you just do not know which one it will be. Tomorrow either rains or it does not, and that is two items.

0:55The match ends in a win, a loss, or a draw, and that is three items - the draw being the one people leave off. The school draw has one slip for every student in the school, so the list is the whole roll. Knowing the list but not the item is the exact situation this subject was built for. It is also the situation in which people are most confident, and most often wrong.

1:25There is an argument that sounds right and is not, and it is worth killing early. It goes: it will either rain or it will not, so the chance is a half. Try that argument on the school draw. It says: either my name comes out or it does not, so my chance is a half. In a school of five hundred, the truth is one in five hundred. The argument is two hundred and fifty times too big.

1:53And notice what it did on the way to being wrong. It never looked at the draw at all. It hands the same answer to every question you will ever put to it. An answer that does not depend on the question is not an answer. The strange part is that the rule buried inside that argument is a real rule. One divided by the number of things that could happen is exactly right - when there is no reason to prefer any one of them.

2:26Five hundred slips, folded the same way, mixed in the same box: one in five hundred. So what failed was not the division. It was the assumption underneath it. Two outcomes written side by side does not make those two outcomes equal. Rain and no rain are not two folded slips in a box, and nothing about listing them together makes them so. The list itself has to be right as well.

2:57Say the match ends in a win or a loss, and the rule gives a half. Say it ends in a win, a loss, or a draw, and the rule gives a third. Losing one outcome out of three has made the answer half again as large as it should be. Fifty per cent too big, from one missing word. That is not a rounding slip. And the damage is worst on short lists, which is exactly where this subject starts.

3:27So here is the move. Instead of guessing what will happen, measure how likely it is. You already know what measuring looks like. Length has a scale and a procedure - lay the tape down, read the number off. Area has one; volume has one. Likelihood gets a scale too, and it runs from zero to one. Zero is the end where the thing cannot happen; one is the end where it must.

3:56Everything genuinely in doubt lives strictly between those ends, and the whole job is to say where. Before any numbers at all, five words will do. Impossible sits at zero, on its own. Certain sits at one, on its own. Equally likely sits at the exact middle, on its own. Less likely is the whole stretch below the middle, and more likely is the whole stretch above it. Five words, five stretches of one scale.

4:28Check them and you find they leave no gap anywhere and never once overlap. Every point on the scale gets exactly one of them, which is already far more than a vague sense of things. But a scale on its own does not make something a measurement, so here is the demand we are going to place on it. Two people, given the same evidence and following the same procedure, should arrive at the same number.

4:59That is what a tape measure gives you. Hand the same desk to two people and their readings differ by a millimetre out of nearly fifteen hundred, and no more than that. What holds them together is the procedure, not any agreement about desks. So: does putting a number on rain do the same? Rather than assume, let us count. Two friends look at the same sky. One sees the sunshine and says rain is less likely than not.

5:31The other feels the heat and says rain is far from ruled out. Neither is wrong, and neither is being careless. Cut the scale into three thousand steps and ask how many of those numbers both readings still allow. Seven hundred and fifty of them. A quarter of the entire scale is left standing after both friends have spoken. And it is looser than even that suggests. Harden one friend's wording from at least a quarter to more than a quarter, and the answer shifts.

6:05The words moved it, not the sky. Now put the school draw through the same test. One slip each, folded the same, one pulled out blind. How many numbers on that same scale of three thousand steps are consistent with it? Exactly one. Not a range, not a leaning - one number, and everybody who understands the setup arrives at it. Seven hundred and fifty against one. That is the distance between describing a feeling and taking a measurement, and closing it is what the rest of this subject is for.

6:45One objection deserves an answer. Perhaps the two friends are simply using different units, the way a length can be given in centimetres or in inches. But a change of unit is reversible. Tell me the centimetres and I can work out the inches; tell me the inches and I get back the centimetres I started with, exactly. Each number determines the other, always, through one fixed factor. Now take two people who agree that the match is a coin toss and disagree completely about the rain.

7:22There is no factor anywhere that carries one person's numbers to the other's. They are not one measurement expressed in two units. They are not a measurement. Last, notice that the two ends of the scale are reached by different roads. Monday follows Sunday: certain, because of how the week was built, and you can check it across seventy-three thousand days in a row without once looking out of a window.

7:53Snow in Singapore in July: as good as impossible, but that rests on what the world does, and no amount of arithmetic will settle it. Same end of the scale, entirely different reason for being there. Now take four ordinary events: that Monday, that snow, an elephant walking through the classroom, and greeting a friend at school tomorrow. They use four of our five words. Equally likely never gets used at all, and that is the clue to what is still missing.

8:27We have a scale, we have names on it, and we still have no way to produce a number that everybody gets. There are two such ways, and they are what comes next.