The impossible and the certain pin the scale at 0 and at 1

0:00Throw a die and ask for an 8. Go through the method exactly as before. List the outcomes: one, two, three, four, five, six. Now cross off the ones that do not make the event happen. One is not an 8. Two is not an 8. Three, four, five, six. None of them is an 8. Everything is crossed off. Nothing is left. This is not a broken question. It is a good question with a good answer, and the method hands it over without complaint.

0:36The favourable count is nought. The total is six. Nought over six. Which is nought. Now be careful about what that number is reporting. It is not saying the 8 is very unlikely, or unlucky, or nearly impossible. It is saying something much flatter than that: of the six things this experiment can end in, the number that are an 8 is nought. The probability is nought because the numerator is nought, and the numerator is nought because the list is.

1:10That is the whole of what the answer means. Now go the other way with the same die. Ask for a result below 7. Same list. One, two, three, four, five, six. Cross off the ones that fail. One is below 7, so it stays. Two stays. Three, four, five and six all stay. Nothing was crossed off at all. The favourable list is the whole list. And notice that we did not decide this by being confident about dice. We decided it by going along the list and finding no exceptions.

1:50Six favourable, six altogether. Six over six. Which is one. Again, read it flatly. It does not say this is as likely as things get. It says the favourable list and the full list are the same list. Every outcome the experiment has qualifies, so there is no room left for the event to fail in. One is what the ratio does when the top and the bottom count the same things.

2:20These two have names, and the names are worth having because they point at the event rather than at the number. An event no outcome favours is called an impossible event, and its probability is nought. An event every outcome favours is called a sure event. It is also called a certain event; the two words mean the same thing here. Its probability is one. Both names describe the LIST, not the feeling. Impossible means nothing on the list does it. Sure means everything on the list does.

2:57And a quick check on that: across every event you can build on every experiment with up to eight outcomes - five hundred and ten of them - the probability is nought exactly eight times and one exactly eight times. Once each per experiment. The empty event, and the whole of it. The other four hundred and ninety-four are strictly in between. So can a probability be anything else? Bigger than one, say, or negative?

3:30Here is the argument, and it is one sentence long. The favourable outcomes are drawn from the possible ones. That is it. The numerator is not an independent number that happens to sit above the denominator. It is a count of part of the very list the denominator counts all of. Watch it climb. Start with an event nothing favours: nought over six. Add one outcome to it: one over six. Another: two over six. Three, four, five.

4:04And then you run out of list. Six over six, and there is nothing left to add. Seven outcomes out of six is not a large probability. It is not a probability at all, because there is no seventh outcome to be favourable. The climb starts at nought, ends at one, and goes up by exactly one outcome's worth at every step. It never steps down and it cannot walk off either end.

4:34A negative one is even quicker to dismiss. Both numbers are counts of things. Neither of them can be less than nothing. Which turns the range into a fact about counting, and that is how to read it. Nought and one are not the ends of a confidence scale that someone chose. They are the two places where the numerator runs out of room - at the bottom of the list and at the top of it.

5:03So a number outside the range is never reporting an unusually unlikely event. It is reporting a broken computation. One point two means somebody counted favourable outcomes that were not in the outcome list. But here is the part worth being honest about. A broken computation does not usually announce itself. Take every way of naming a set of things when only some of them are outcomes of the experiment - two hundred and forty-eight of them, on experiments with up to five outcomes.

5:41A hundred and eighty-six of those name something the experiment cannot produce. Only thirty come back with a number above one. The other hundred and fifty-six hand you a perfectly respectable answer between nought and one. So a number outside the range proves the count was wrong. A number inside it proves nothing at all. Checking your list is not optional. It would be easy to think nought and one are things that happen to small experiments. They are not.

6:17Throw two dice and add them. Thirty-six ordered pairs, laid out as a six by six grid. Ask for a sum of 13. Go along the grid and shade every cell that qualifies. One and one is two. Six and six is twelve. The largest sum anywhere on the board is twelve. Nothing gets shaded. Nought out of thirty-six, which is nought. Now ask for a sum of at most 12.

6:46Every cell qualifies, because twelve is the largest sum there is. The whole grid shades. Thirty-six over thirty-six. One. Same two answers, on a board thirty-six times bigger, reached by shading nothing and by shading everything. And in between them the board is full of ordinary answers. A sum of seven is six cells out of thirty-six, which is a sixth. Now a question in the other direction. You are handed four numbers and asked which of them could be a probability.

7:23Two thirds. Minus one point five. Fifteen per cent. And nought point seven. Take them one at a time, and do not just check the range - find an experiment that delivers the number, because that is what being a probability means. Two thirds. Two outcomes out of three. An experiment with three outcomes and an event holding two of them does it. Fifteen per cent. Fifteen per cent is nought point one five, which is three twentieths. Three outcomes out of twenty.

7:59Nought point seven is seven tenths. Seven outcomes out of ten. And minus one point five? There is no count of favourable outcomes that is negative, and no total that is either. Nothing delivers it. So the answer is the negative one, and it is the only one. The trap here is the percentage. A probability is a number, not a notation, and fifteen per cent is the same number as three twentieths written a different way.

8:32Rejecting it because it does not look like a fraction is rejecting the handwriting instead of the value. One experiment can hand you both ends at once. A bag holds candies, and every single one of them is lemon flavoured. Take one out without looking. What is the chance it is orange flavoured? No candy in the bag is orange, so the favourable count is nought. The probability is nought.

9:03What is the chance it is lemon flavoured? Every candy is, so the favourable count is the whole bag. The probability is one. And notice what was never needed: the number of candies. Try it with a bag of one, or of twelve, or of any size in between, and the two answers do not move. Nought over n is nought and n over n is one, whatever n is.

9:30One more, with an ordinary answer beside the extreme one. A spinner with eight equal sectors, numbered one to eight. The chance of an 8 is one sector out of eight. An eighth. That is an ordinary answer, sitting comfortably inside the range. The chance of a number below 9 is all eight sectors out of eight. One. Same spinner, same list, two questions - and one of them happens to land on the end of the scale.

10:03So, the whole thing. Every probability sits between nought and one, endpoints included, and not because anyone decided the scale should look like that. The favourable outcomes are drawn from the possible ones, so the numerator starts at nothing and can climb no further than the denominator. Nought is the bottom of that climb: an event no outcome favours. One is the top: an event every outcome favours. Those are the impossible and the sure event, and both names are about the list rather than about how anybody feels.

10:43The practical value of knowing this is that it makes an answer checkable. If a computation ever hands you a probability above one or below nought, you do not need to think about whether the event is strange. You need to go back and look at what you counted. And if it hands you something inside the range, that is not evidence that you counted correctly. It only means the mistake, if there is one, is still hiding.