When outcomes cannot be counted, measuring length or area instead

0:00Every experiment so far has had a list you could write down. A coin: two results. A die: six. Two dice: thirty-six ordered pairs. A pack of cards: fifty-two. Long lists, some of them, but finite. You could in principle write out every entry and stop. That mattered more than it looked, because the rule you have been using needs it. Favourable over total. Both of those are counts, and a count needs something finite to count.

0:32Now here are two experiments where there is no such list at all. The first. A piece of music runs for two minutes, and it is stopped at some instant while it plays. The result of that experiment is a time. Not one of a few times. Any time at all between nought and two. One minute. One and a half. One point three three three, and on forever. Between any two instants you name, there is another one.

1:03The second. Something comes down somewhere inside a rectangular patch of ground, and you want to know where it landed. The result is a point in the plane. Again, no list. Between any two points there is another point, and there is no first one and no next one. So try the rule on the music anyway, and watch it jam. The bottom number. How many results can the experiment produce?

1:33Infinitely many. Not a large number - no number at all. The top number. How many of them lie in the first half-minute? Infinitely many again. Infinity over infinity is not a quantity. It is not a large answer, or a small one, or zero. It is nothing. The rule, exactly as you learnt it, cannot be applied here. And that is a real gap, not a technicality. Stopping a piece of music is not an exotic experiment.

2:08So the useful move is to ask what that ratio was measuring in the first place. It was never really about counting. Counting was the method, not the meaning. What it measured was a share. How much of everything that could happen goes your way. On a finite list, taking a share is easy. Everything is one outcome wide, so you count the ones you want and divide by how many there are.

2:36But the share is the idea, and the counting is only the arithmetic. And a share survives when the list does not. If the results fill a stretch of time, the share is a share of that stretch. A length over a length. If they fill a patch of ground, it is a share of that patch. An area over an area. Nothing new has been assumed. The same idea has been measured with a different instrument.

3:06Back to the music, with length doing the work. Draw the two minutes as a stretch of line, from nought to two. Every instant the music could stop at is a point somewhere on that line, and none is favoured over another. The whole experiment measures two. The question was: does it stop in the first half-minute? So shade from nought to a half. That stretch measures a half. The share is a half over two.

3:35Which is a quarter. And notice what did not happen. Nothing was divided by infinity. Two numbers were measured and one was divided by the other. Two things worth pinning down before moving on. First, a half-minute at the far end - from one and a half to two - measures a half as well, so it is also a quarter. Same length, different place, same answer. Hold on to that; it comes back.

4:04Second, the awkward one. What is the chance the music stops at exactly one minute? One minute is inside the interval. It is a perfectly possible result. But a single instant has no width. It measures nothing at all, so the share is nought. That is a boundary of something you already know. On a finite list, a chance of nought meant the event was empty - there was no outcome in it.

4:34Here the event is not empty, and the chance is still nought. The old reading stops being safe the moment the list does. Now the second experiment, where the results fill a patch of ground rather than a stretch of time. A rectangular region, nine kilometres along the bottom and four point five up the side. Nine times four point five is forty point five square kilometres. That is the whole experiment.

5:06Somewhere inside it there is a lake, and something has come down at a point somewhere in the region. The chance it came down in the lake is the lake's area over forty point five. Every step of that is the same argument as the music. Only the instrument changed, from length to area. So the whole question is now: how big is the lake? And that is where the figure lays a trap, so read it slowly.

5:36Four measurements are marked on it. Nine along the bottom. Four point five up the left side. Those two give the region, and we have used them. Then six along the top. And two up the right side. It is very tempting to take those as the lake: three by two. Look at where the arrows actually run. The six starts at the left edge and stops where the lake begins. It measures the ground beside the lake, not the lake.

6:07The two starts at the bottom edge and stops where the lake begins. It measures the strip below the lake. Neither of them touches the lake at all. The lake's own sides are nowhere marked, and you have to recover them. Across: nine take away six is three. Up: four point five take away two is two point five. So the lake is three by two point five, which is seven point five square kilometres.

6:36Seven point five over forty point five. Five twenty-sevenths. Read the arrows as the lake and you get three by two, six square kilometres, four twenty-sevenths - out by one twenty-seventh, and with no way of telling. Now a claim worth testing, because it is the best evidence that area really has taken over the job the count used to do. Take that same lake - three by two point five - and slide it.

7:08Put it in the far corner. Five twenty-sevenths. Put it hard against the left edge. Five twenty-sevenths. Put it in the middle. Five twenty-sevenths. Nothing moved, because neither area moved. The answer never depended on where the lake was. Now change its size instead. Make it three by two. Four twenty-sevenths. The answer moves at once. Size matters and position does not, which is exactly what you would expect of a measure and would not expect of a picture.

7:44One thing did get assumed along the way, and it is worth saying out loud. On a finite list the assumption was that the results are equally likely. Here it becomes: the results are spread evenly over the region. That is not automatic, and when it fails the length rule gives the wrong answer. Suppose the music is far likelier to be stopped early - say all the weight sits in the first minute.

8:13The first half-minute is still a half of the two minutes, so the length rule still says a quarter. The true share is a half. The rule is out by a factor of two. Take a hundred and eighty-nine combinations of stretch and spread and check them all. The length rule is right in a hundred and twenty-nine of them, and wrong in sixty. Fifty-four of them are evenly spread, and it is right in every single one. Not one even case fails.

8:44Position stops being irrelevant too. Put that same lake on ground where one end is weighted, slide it through the same three places, and the three answers are three different numbers. So evenly spread is the licence, exactly as at random and well shuffled were. One more, and it ties the two instruments together. A rectangle three metres by two, so six square metres. Inside it, a circle of diameter one metre.

9:18A die is dropped at random onto the rectangle. What is the chance it lands inside the circle? First, the die is being treated as landing at a point. That is an idealisation, and it is worth naming: a real cube covers ground, and this argument ignores that. Now the circle. Diameter one, so the radius is a half. That step is where the marks go. Area is pi times the radius squared. Pi times a half squared is pi over four square metres.

9:54The share is pi over four, over six. Which is pi over twenty-four. About nought point one three one. Leave it as pi over twenty-four; that is the exact answer. Use the diameter where the radius belongs and you get four times too much, which is the commonest way to lose this one. And where the circle sits in the rectangle makes no difference at all - which by now should be the thing you expected.

10:24The definition never changed. Favourable over possible, all the way through. What changed is what you measure them with.