The 0-to-1 scale, and what the endpoints mean

0:00Start with a number. Somebody tells you the chance of winning is nought point seven five. What exactly has been claimed? Three things at once, and they are all the same thing. Seventy-five out of every hundred. Three out of four. And a position three quarters of the way along a line running from nought to one. That last version is the one worth carrying, because it puts the number somewhere.

0:29It sits past the middle, so winning is more likely than losing - and short of the end, so winning is not guaranteed. All of this is about that line. Now the middle of it. A half - nought point five, or fifty out of a hundred. There is a reading of that number which is extremely common and simply wrong: that a half means you have no idea. It does not.

0:57A half is a claim about the event, that the two outcomes are equally likely and neither has any edge. Not knowing is a state of the person, and it carries no number at all. If you genuinely have no idea, you have not measured anything - and a half is a measurement like any other. Then the two ends of the line. Nought means the event cannot happen. Not unlikely - cannot.

1:26Winning a match you never played is nought. One means it must happen. Reach into a bag where every sweet is purple and you come out with a purple sweet, because there is nothing else to come out with. Notice that neither end is about history. The bag is certain because of what is in it, not because of what happened last time. And most events sit strictly between the two - the ends are real values, and they are the exceptions.

1:58So why those two numbers, and not minus five to a hundred? Nobody chose them. Look at how a probability is built: count the outcomes that qualify, and divide by all of them. A part, over the whole it is a part of. That fraction cannot come out negative, because you cannot count fewer than none of them. And it cannot come out larger than one, because you cannot count more of them than there are.

2:27Sweep every deck you like, of every size, and not one escapes at either end. The two ends are not a rule to be remembered; they are what counting a part of a whole does. Here is a machine that shows it. Six cards, face down, some purple and the rest green. Draw one without looking, and the question is the chance of purple. Six cards, always - that number does not move.

2:57The only thing changing is how many of them are purple. That restriction matters more than it sounds. Change the deck size and the purple count together and you have learned nothing, because you cannot tell which one moved the answer. One purple card in six is one thing; one purple card in fifty-two is quite another. So hold the size still, and turn the one dial. Turn it. No purple at all: nought out of six, which is nought, and that deck sits on the left-hand end.

3:34One purple: one out of six, a little way up. Three purple: three out of six, which is exactly a half, the middle. Five purple: five out of six, well up the scale. And six purple: six out of six, which is one - you cannot fail to draw purple, because there is nothing else in the deck. Five decks, five positions, and the scale drawn out by nothing more than counting.

4:02But five is not all of them. Six cards give seven decks, not five, and the two usually left out are two purple and four purple - worth a third and two thirds. Put them back and something the five were hiding becomes visible. The seven sit exactly one sixth apart, an even ladder from one end to the other. With only five, the gaps run one rung, two, two, one, and the scale looks as though it moves in jumps.

4:33It does not; it moves in sixths, and every step is the same size. The ladder also shows what six cards cannot reach. Nought point seven five, the number we opened with, is not on it - it falls between two thirds and five sixths. For exactly three quarters you need a deck whose size divides by four; thirty-nine cards out of fifty-two would do it. There is a trap here worth naming.

5:03Draw five labels evenly spread across the scale, sit the five decks underneath them, and it becomes very natural to read the label positions as the values. Spread evenly from nought to one, five labels sit at nought, a quarter, a half, three quarters, and one. The decks are actually at nought, one sixth, a half, five sixths, and one. Three of the five agree. Two do not - and both are wrong by exactly one twelfth, one too high and the other too low.

5:38Equal and opposite, which is precisely why the mistake looks tidy and goes straight past you. The words on the scale need care too, because they are not all the same kind of thing. Impossible is a single number: nought. Even chance is a single number: a half. Certain is a single number: one. But less likely and more likely are not numbers at all - they are stretches of the line.

6:08Of the seven rungs on our ladder, one is impossible, one is an even chance and one is certain. The other four live in the two stretches, two on each side. Three of the words pin a value; two mark a region, and reading all five as values is the mistake. Put five events on the line and see where they fall. A die showing a number above six: none of its six faces qualify, so nought - impossible.

6:39A three on one roll of that die: one face out of the six. Heads on one flip of a coin: one out of two, the middle exactly. A card from two to ten drawn out of a full pack: nine values in each of four suits, so thirty-six of the fifty-two. And a purple sweet from a bag where every sweet is purple: one. Look hard at the fourth one.

7:06It is more likely than not - and it is still smaller than five purple cards out of six. More likely is a stretch, and things inside it are not all the same. A word about the numbers themselves. Three quarters, nought point seven five, and seventy-five per cent are not three facts. They are one number written three ways, and you should be able to slide between them without stopping to think.

7:35But be careful with the sliding. One sixth written as nought point one six seven is not one sixth. It is larger, by one part in three thousand. For most purposes that will not matter; for arithmetic later on, it will. Of our seven rungs, only three come out as whole percentages - nought, fifty and a hundred. The fraction is the exact form, and the decimal is the convenient one.

8:07Finally, the shorthand. The probability of an event E is written P of E, and everything in this video compresses into a single line. Nought is less than or equal to P of E, which is less than or equal to one. Both of those signs are earned rather than declared. The left one, because you cannot count fewer than none of the outcomes that qualify. The right one, because you cannot count more of them than exist.

8:38And both are less-than-or-equal rather than strictly less, because the two ends really are reachable - the empty deck and the full one are both decks. One line, and it is a summary of the counting rather than a rule laid on top of it. What it does not yet tell you is how to get the number when the outcomes are not equally likely, and that is what comes next.