The two extreme cases, and the difference between one outcome and many

0:00An event is any subset of the list of outcomes. Any. That word was not chosen for elegance, and it is not free. It admits collections nobody set out to build, and two of them arrive the moment the definition is written down. One of them holds no outcome at all. The other holds every outcome there is. Neither was invented to fill a gap in a list of names.

0:28This video is about where those two come from, and about a second sorting that has nothing to do with them. So begin with a die. One throw of an ordinary die, and six outcomes: one, two, three, four, five, six. Here is a description of a result. The face that comes up is a multiple of seven. Walk it past the six faces and ask which of them answer to it.

0:57One is not a multiple of seven, nor is two, nor three, nor four, nor five, nor six. The smallest positive multiple of seven is seven itself, and seven is already past the last face on the die. So the description picks out nothing. Notice what has not gone wrong. That is a perfectly grammatical sentence about a perfectly ordinary experiment. It names a collection of outcomes, the collection turns out to be empty, and an empty collection is still a subset.

1:33The definition takes it without complaint, and its size is nought. Same die, and a second description. The face that comes up is odd, or else it is even. One is odd, two is even, three is odd, four is even, five is odd, six is even. Every face answers to it, and nothing at all is left out. So this description picks out the whole list, and its size is six.

2:03One die, two sentences, and they have landed at opposite ends of everything the definition allows. Nothing between them was manufactured, and neither was. You should not take my word for it that those two sentences were ordinary. I chose them, and someone who wanted the ends to appear would choose exactly such sentences. So let nobody choose. Build every sentence of one very plain shape instead. The face that comes up is, then a comparison, then a number.

2:39Five comparisons: exactly, at least, at most, more than, fewer than. And every number from nought up to seven, which is eight of them. Five comparisons times eight numbers is forty sentences, and no judgement went into the list. Now run all forty past the six faces and see where they land. Eight of the forty name the collection holding nothing. Six of them name the whole list. So fourteen of the forty arrive at one end or the other, and the remaining twenty-six land somewhere in between.

3:17Between them the forty reach sixteen different collections, out of the sixty-four this die has altogether. The two ends are not rare and they are not exotic. They are where a third of the plainest sentences you can write about a die end up. Suppose you disliked them anyway, and ruled that those two collections are not events. Here is the bill for that. Take those same forty ordinary events and pair them off, every one with every other, which makes seven hundred and eighty pairs.

3:53Ask of each pair what the two have in common. Four hundred and nineteen of those pairs have nothing in common at all. Ask instead what the two cover between them. Two hundred and eighty-three of the pairs cover the whole list. So the overlap of two ordinary events lands on the empty collection constantly, and the union of two ordinary events lands on the whole list constantly. A definition that struck the two ends off would have to say that the overlap of two events is sometimes not an event.

4:31That is a far stranger sentence than the one it was trying to avoid. Admit every subset, and the two ends come in with the rest, and nothing has to be said twice. They have names. The event holding no outcome is the impossible event. No result of the experiment makes it occur, because occurrence is membership, and nothing is a member of it. The event holding every outcome is the sure event.

5:02Every result makes it occur, because every outcome is a member of it. Check that on the die: nought of the six faces make the impossible event occur, and all six make the sure event occur. Check it on three tossed coins, which have eight outcomes: nought of the eight, and all eight. The two ends are not a peculiarity of one experiment. They exist wherever there is an experiment, and they are the same two sentences about membership every time.

5:37That was one way of sorting events, and it looks at where an event sits: at one end, at the other, or somewhere in the middle. There is a second way, and it looks at something else entirely. It asks how many outcomes the event holds. Call that number the size of the event. The impossible event has size nought and the sure event on a die has size six, so the two sortings are not the same question asked twice.

6:07And one size in particular is worth a name of its own. Size one. Two tosses of a coin, four outcomes: head-head, head-tail, tail-head, tail-tail. Take the description: the result is head-head. The collection it names holds exactly one outcome, and written out it is a pair of braces with head-head inside. An event of size one is called a simple event. It is also called an elementary event; the two names mean the same thing.

6:42And there is something here worth slowing down for. Head-head and the set holding head-head are not the same object. One of them is an entry on the list of outcomes. The other is a subset of that list. The set is an event; the outcome is not, because an event is a subset and an outcome is a member. That is true of all four: all four of those sets are events of this experiment, and none of the four outcomes is.

7:13Later on you will see them written alike for convenience, and convenience is all that is. Four outcomes, and four simple events, one for each. That equality is not a coincidence to be checked experiment by experiment, and here is why. Pair each outcome with the simple event that holds it. Head-head with the set holding head-head, and so on down the list. Three things have to be true for that pairing to force the counts equal, and all three are.

7:48No outcome is left without a partner. No simple event is left without a partner. And no outcome is sent to two different partners. If the pairing failed you would see it: stop it one outcome short and one outcome is left over and one simple event is left over, and it is tail-tail on both sides. With the pairing complete there is nothing over on either side, so four outcomes give four simple events and cannot give any other number.

8:21Grow the experiment and the pairing grows with it: two and two, four and four, eight and eight, sixteen and sixteen. Run it out to every list up to twelve outcomes and the two counts never once drift apart. It is very easy to slide from there into thinking there are as many events as outcomes. There are not, and the gap is enormous. Four outcomes carry sixteen events altogether, and four of those sixteen hold a single point.

8:54Eight outcomes carry two hundred and fifty-six events, and eight of those hold a single point. So the count of events is two to the power of the count of outcomes, against the count of outcomes itself. Two to the n beats n at every size, and it beats it by more at each step. Which means the simple events are the rare ones, and they get rarer. Four in sixteen is a quarter of them; eight in two hundred and fifty-six is one in thirty-two.

9:27Follow that share from two outcomes out to twelve and it falls at every single step. Simple events are the smallest kind of event, and they are also the scarcest. Now take three coins, tossed together, with all three recorded in order. Eight outcomes: three heads; head, head, tail; head, tail, head; head, tail, tail; tail, head, head; tail, head, tail; tail, tail, head; and three tails. Three descriptions, and the collection each one names.

10:01Exactly one head. Three outcomes answer to it: the ones with a single head, wherever that head sits. At least one head. Seven outcomes answer to it, and the single outcome it leaves out is the one with no head at all. At most one head. Four outcomes answer to it: the three with a single head, together with that same all-tails one. So the three sizes are three, seven and four.

10:31Every one of them holds more than one outcome. An event of size greater than one is called a compound event, and all three of these are compound. There is a tempting reading of that word, and it is wrong. Compound sounds like it means the description was built by joining two things together with an and or an or. None of those three descriptions was. Each one is a single short phrase, and all three came out compound.

11:04But that is an argument from three examples, so test it properly: keep the words fixed and move the experiment. Ask the same three questions of a coin tossed once. Exactly one head now picks out a single outcome. At least one head picks out that same single outcome. At most one head picks out two. So on a single toss, only one of the three is compound, and the other two are simple events.

11:34Run the identical three sentences over one toss, two tosses, three, and on up to ten. The number of them that come out compound is one, and then three, nine times over. Not one word of any sentence changed across those ten runs. So whatever decides compound, it is not the phrasing. It is the size, and the size belongs to the experiment. Those three events on three coins are worth one more look, because they are not sitting side by side.

12:09The exactly-one event sits inside the at-least-one event, and it is smaller, so that containment is strict. It sits inside the at-most-one event too, and is smaller than that one as well. Neither of the bigger two sits inside the other. So ask what those two have in common. The at-least-one event and the at-most-one event share exactly three outcomes, and those three are precisely the exactly-one event. The overlap of the two is the third one, on the nose.

12:45And between them they cover the whole list, because every outcome either has a head or has none. Which settles something that sounds obvious and is false. At least one head and at most one head are not opposites. Opposites share nothing, and these two share three outcomes. The real opposite of at least one head is no head at all: it shares nothing with it, and the two together cover everything.

13:14So there are now three names in play. The two ends, the impossible event and the sure event. The simple events, of size one. And the compound events, of size greater than one. Three names look like a sorting, so let us check whether they are one. Take all two hundred and fifty-six events on eight outcomes and ask of each which of the three names it answers to. First question: is anything missed?

13:44Nought of the two hundred and fifty-six escape all three names. In particular the empty one is not missed at all: the first name holds it outright, since the impossible event is exactly that collection. Second question: does anything answer to two names at once? Exactly one event does. It is the whole list, which is the sure event and therefore one of the two ends, and which also holds more than one outcome and is therefore compound.

14:17That double naming happens at every size of experiment, not just this one. And which name it doubles with moves: on an experiment with a single outcome, the whole list holds exactly one point, so the sure event is a simple event there. So the three names cover everything, and they are not disjoint. That is not a flaw, but it is worth knowing, because a list of three names invites you to believe each event has exactly one of them.

14:51Now put the first name aside and keep only the two that talk about size. Simple, and compound. Search all two hundred and fifty-six events for one that neither of those reaches. Exactly one turns up. It is the collection holding nothing. Its size is nought, and nought is neither one nor more than one, so neither size name has anything to say about it. The same single gap opens on every experiment, whatever its size.

15:24Which is worth being careful about, because it is easy to overstate. The empty event is not unclassified. It has a name, and the name is the impossible event. What is true is narrower: sorting by size alone does not reach it, and you need the first name to catch it. One last property, and it is the one everything after this depends on. Take the eight simple events on the three coins and compare them two at a time, which is twenty-eight pairs.

15:59Nought of those twenty-eight pairs share an outcome. Different simple events never overlap, because each holds a single outcome and they are different outcomes. Which means no two of them can occur together, ever. And now the other half of it. Take any one of the two hundred and fifty-six events and build it back up from the simple events of its own members. Every single one of the two hundred and fifty-six comes back exactly.

16:31So an event is its outcomes and nothing besides, and its outcomes come in pieces that never overlap. That is a list of separate parts covering the whole event, which is precisely what you need if you ever want to add something up across an event, one outcome at a time. Two sortings, then, and they answer different questions. The first asks where an event sits, and it hands you the two ends: the impossible event and the sure event.

17:04Neither was invented: fourteen of forty plain sentences about a die land on one or the other, and ordinary events overlap into the first and cover into the second constantly. The second sorting asks how big an event is, and it separates the simple events from the compound ones. Size one is not a small idea. There are exactly as many simple events as outcomes, forced by a pairing with nothing left over on either side.

17:38They are the rarest kind of event, four in sixteen and eight in two hundred and fifty-six. They never occur together. And every event whatever is built out of them. The names sort events; the sizes do the work.