Every description of a happening picks out a part of the outcome list

0:00Toss one coin twice, and write down what both tosses showed, in order. There are four things that can happen. Head then head. Head then tail. Tail then head. And tail then tail. Those two in the middle are not the same thing. Both of them show a single head, and they differ in which toss produced it, so they are two entries on the list and not one. Merge them and there are three entries instead of four, and every count in the rest of this video moves.

0:34The list is the starting point, and it has one property that everything else rests on. Exactly one of those four is going to happen. Not two of them, and not none of them. Now here is a sentence. Exactly one head. That is a description of a happening, written in ordinary words, and the list has no idea what to do with it yet. So take the sentence to each outcome in turn and ask a single question: is this one of the outcomes that makes the sentence true?

1:10Head then head has two heads, so no. Head then tail has one, so yes. Tail then head has one, so yes. Tail then tail has none, so no. Two of the four answered yes, and the sentence has finished its work. What it has left behind is a collection of outcomes. Head-tail, and tail-head. One sentence is not enough to see the pattern, so here are six, all asked of the same four outcomes.

1:41The number of tails is exactly two. Only the last outcome has two tails, so that sentence picks out one outcome. The number of tails is at least one. That is true of head-tail, tail-head and tail-tail, and false only of the outcome with two heads. Three outcomes. The number of heads is at most one. Head-head is out, and the other three are in. Three outcomes again. Hold on to that, because we are going to come back to it.

2:16The second toss is not a head. That one does not count anything at all; it reads a single position and asks what is sitting in it. Head-tail and tail-tail, so two outcomes. The number of tails is at most two. Two tosses cannot produce more than two tails, so every outcome qualifies, and that sentence picks out all four. The number of tails is more than two. No outcome qualifies, and that sentence picks out nothing whatever.

2:49Six sentences. The sizes of what they picked out run one, three, three, two, four and nought. Look at the second and the third. At least one tail. At most one head. They are talking about different faces of the coin. One of them puts a floor under a count and the other puts a ceiling on a different count. And they picked out the identical three outcomes. Not three outcomes that happen to be the same number of outcomes.

3:21The same three. Check it one outcome at a time and they agree four times out of four. The reason is easy to see once you look at what each of them throws away. Both of them exclude head-head, and neither of them excludes anything else. So the collection of outcomes that survives is the same collection, and two quite different sentences have arrived at one place. That is the moment where it is very easy to learn the wrong lesson.

3:54It looks like a fact about English. At least one tail means at most one head, so those two phrases are interchangeable. They are not. Take the same two sentences and ask them of a coin tossed three times instead. At least one tail is now true of every outcome except the one with three heads, so it picks out seven of the eight. At most one head is true of only four of them.

4:22Seven against four. The two sentences have come apart. Run them over one toss, two tosses, three, and on up to eight, and they agree on exactly one of those eight experiments. The one with two tosses. So the coincidence we just watched is not a property of the words. It is a property of this experiment, and the words did not know it was coming. The last two of the six sentences are worth stopping on, because they are the ones people want to throw out.

4:58One of them picked out nothing at all. The other picked out everything there is. Both of those are collections of outcomes from the list, exactly like the other four, and there is no principled place to draw a line that keeps the middle four and refuses these two. And it is not a quirk of coins. Throw one die and ask for a number below seven: every face qualifies.

5:26Ask for a number above seven: none of them does. Same two ends, different experiment. A description that nothing satisfies still names something, and what it names is the collection with nothing in it. So here is the definition the whole subject is built on. An event is any subset of the list of outcomes. Any. Not any subset that somebody has a phrase for. Not any subset that is easy to say.

5:58Any subset at all, including the one with everything in it and the one with nothing in it. That word is doing real work, and the rest of this video is about how much. Because the natural thing to want instead is a definition built out of descriptions: an event is a happening you can describe, and the outcomes are just how you check it. That sounds harmless. It is not, and the way to see why is to count.

6:30Four outcomes. How many collections of them are there? Take them one at a time. There is the empty collection. There are four collections holding a single outcome. Six holding a pair. Four holding three. And one holding all four. Add those up and you get sixteen. Sixteen events sitting on four outcomes, whether or not anybody ever writes a sentence about them. Our six sentences reached five of them, because two of the six landed together.

7:05Five out of sixteen. Eleven of those events never appeared on the board at all, and nothing was wrong with the six sentences. There are simply more collections than there were sentences. Six sentences is a small sample, so let us be fairer to language than that. Build every sentence of one very ordinary shape. The number of some face is in some relation to some number. Two faces to choose from, heads or tails.

7:38Five relations: exactly, at least, at most, more than, fewer than. And three numbers, nought, one or two, because two tosses cannot give you more than two of anything. Two times five times three is thirty sentences, and nobody chose which ones to include; they were generated. Thirty sentences, asked of four outcomes. How many different collections do they reach between them? Seven. Thirty sentences, seven events. They fall into seven groups, and the biggest group has six different sentences in it, all of them naming one collection.

8:20Seven out of sixteen, so nine of the events are out of reach of that whole family. And one of the nine is not exotic at all. The two tosses agree. That is a perfectly ordinary thing to want to say, and it picks out two outcomes: head-head, and tail-tail. It is a collection of outcomes like any other. But not one of those thirty sentences can name it. The reason is worth saying slowly.

8:50Every one of them reads only how many heads there are, and head-tail and tail-head have the same number of heads. A sentence that can only count is blind to the difference between them. Check that directly and not one of the thirty separates those two outcomes. Which also means neither head-tail on its own, nor tail-head on its own, is reachable by counting. So describability is not what makes an event.

9:20The collection exists; the shape of sentence you brought is what ran out. Now go back to the property we started with. Exactly one outcome happens. That is what makes the whole translation work, because it means every description is settled the instant the outcome is known. There is no waiting, and no description can be left half true. And it works the other way too. Take any two different events on those four outcomes.

9:52There is always an outcome that lies in one of them and not the other, and over every one of the hundred and twenty pairs there is, that holds a hundred and twenty times. No event is ever told apart from itself, and no two different events are ever left indistinguishable. So the collection carries everything that could ever be observed about the description, and the sentence carries nothing beyond the collection.

10:21That is the licence to throw the sentence away. Which brings us to what it means for an event to occur. Throw one die, and let the event be a number below four. As a collection, that is one, two and three. Now throw. If a one comes up, the event occurred, because one is in the collection. A two: occurred. A three: occurred. A four: it did not. Five, no.

10:51Six, no. Three of the six outcomes make it happen and three do not, and notice what did not change across those six sentences. The event. It was the same three numbers every single time. Occurrence is not a property of the event; it is a question asked of the event and one outcome together, and the whole of the question is whether that outcome is inside. Nothing about how likely it was has appeared anywhere in this video, and nothing needs to.

11:27The translation drill is the same on any experiment, so run six descriptions past one die. A number below seven picks out all six faces. A number above seven picks out nothing. A multiple of three picks out three and six. A number below four picks out one, two and three. An even number above four picks out six alone. A number not below three picks out three, four, five and six.

11:57Six, nought, two, three, one and four. Two of those six are the two ends again, and the middle four are ordinary sentences that happen to land on ordinary collections. Make the experiment bigger and the reason for keeping sets rather than sentences becomes hard to argue with. Roll a pair of dice, and record both, so there are thirty-six outcomes. The sum is above eight: ten outcomes. A two appears on one die or the other: eleven outcomes.

12:33Careful with that one. Six outcomes have a two on the first die, six have a two on the second, and one of them has a two on both, so it is eleven and not twelve. The sum is at least seven and also a multiple of three: five outcomes, the ones adding to nine and the one adding to twelve. And now the sets do work that sentences cannot.

12:58The first two cannot both happen, because the largest sum you can reach with a two on a die is eight. Nor can the second and the third. But every outcome of the third is already inside the first, because nine and twelve are both above eight. None of that was visible in the wording. It was visible the moment the wording was traded for a collection. One more count, and it is the one that settles the argument.

13:29Toss three coins and there are eight outcomes. The number of events on eight outcomes is two hundred and fifty-six. Counted by size: one empty, eight holding a single outcome, twenty-eight holding a pair, fifty-six holding three, seventy holding four, and then the same numbers again on the way back down. They add to two hundred and fifty-six exactly. Three heads is one of the eight single-outcome events. A head on the first coin is not; that one holds four outcomes.

14:05Now go back to the pair of dice, with its thirty-six outcomes. The number of events there is more than sixty-eight thousand million. Nobody was ever going to describe those one at a time. A definition that only admitted the describable ones would have had to say which sentences count, in what language, and how long they are allowed to be. So that is what the word any is buying.

14:35A description is how a person reaches an event. It is not what the event is. Two unlike sentences can arrive at one collection, and we watched that happen and then watched it stop happening the moment the experiment changed. A whole family of thirty sentences can leave nine collections unreachable, including one as plain as the two tosses agreeing. And once the sentence is traded for the collection, the operations you already know on collections are waiting: what two events have in common, what falls in either, what is left over.

15:16That is the next thing, and it only works because an event was defined as any subset and not as a sentence. The sentence gets you in. The set is what survives.