Single-outcome events, and why all of them together come to 1

0:00An event is one or more outcomes, taken together. So the smallest event there can possibly be is one outcome on its own. That has a name. An event holding exactly one outcome is called an elementary event. It is not a new object. It is an outcome that has been promoted. Which means an experiment with six outcomes has exactly six elementary events, and one with nine has nine.

0:31One per outcome, always, because that is what one-per-outcome means. And each of them, by the rule from last time, is one favourable outcome out of however many there are. One over n, every time. Before going anywhere, sort the events we have already met into two piles. A coin comes up heads: one outcome. Elementary. A coin comes up tails: one outcome. Elementary. Drawing the yellow ball, the red ball, the blue ball: one outcome each. All three elementary.

1:09Now a die lands above four. That is five, and six. Two outcomes. Not elementary. And a die lands on four or less. One, two, three, four. Four outcomes. Not elementary either. So the test is not how the event is described. It is how many outcomes the description turns out to hold. Anything described by a condition rather than by a single result is a candidate for holding several.

1:42Now add up all the elementary events of one experiment and see what happens. The coin. Two elementary events, a half each. A half plus a half. One. The bag of three balls. Three elementary events, a third each. A third plus a third plus a third. One. Two different experiments, two different numbers of outcomes, and the same total. That is worth stating as a general fact: add up what every elementary event of an experiment carries, and you always get one.

2:18But stating it is not the same as knowing why, and the why is where everything interesting is. Because here is a pair that also totals one, and neither of them is elementary. The die again. Above four is a third. Four or less is two thirds. A third plus two thirds. One. But the first event holds two outcomes and the second holds four. So totalling one is not what makes an event elementary, and being elementary is not what makes a total come to one.

2:54Those are two different things and it is very easy to weld them together by accident. Whatever is really going on has to explain both cases at once: two events of sizes two and four, and six events of size one. So why does it happen? Look at what the fractions are made of. Every one of them has the same bottom number, because the bottom number is the size of the outcome list and there is only one outcome list.

3:27And the top numbers are the sizes of the events. So adding the fractions is adding the sizes, over an unchanged bottom. For the die: two plus four, over six. Six over six. For the six elementary events: one plus one plus one plus one plus one plus one, over six. Six over six. Same total, for the same reason, and the reason has nothing to do with how big the pieces are.

3:58The numerators divide the denominator between them. That is the whole fact. Which immediately tells you what could go wrong. The sizes of the events add up to the size of the whole list only if two things are true, and neither of them is automatic. First: nothing is left out. Every outcome belongs to at least one of the events. Second: nothing is counted twice. No outcome belongs to two of them.

4:29Miss an outcome and the sizes come to less than the whole, so the total falls short of one. Count one twice and the sizes come to more than the whole, so the total overshoots. Get both right and the sizes are exactly the whole, and the total is exactly one. There is a nice way to see it. Walk along the outcomes instead of along the events, and ask each outcome how many of the events contain it.

4:58The answer has to be one, for every single outcome. Not nought, which is a gap. Not two, which is an overlap. A list of events like that is called a partition of the outcomes. You do not need the word. The two conditions are the thing. Now the example that makes it worth the trouble. A box holds three blue marbles, two white ones and four red ones, and you draw one without looking.

5:28How many outcomes? Nine. Not three. Nine marbles, nine ways the draw can come out, and the colours are not the outcomes. Ask about colours and you get three events, of sizes three, two and four. Blue is three of the nine, so a third. White is two of nine. Red is four of nine. Not one of those three is elementary. But every marble is exactly one colour, and every colour is somebody in the box.

5:59Nothing left out, nothing counted twice. A partition. So the total is three plus two plus four over nine. Nine over nine. One. Same nine outcomes, two completely different groupings - nine singles, or three colours - and the same total, because the total was never about the grouping. Now break it, three times, on purpose. Take a standard pack of fifty-two cards. First: a heart, and a spade. Thirteen cards each, sharing nothing.

6:34Thirteen plus thirteen over fifty-two. Twenty-six over fifty-two. A half. They do not overlap, but they leave twenty-six cards out, so the total falls short. Second: a red card, and a card that is not a heart. Twenty-six and thirty-nine. Between them they cover every card in the pack, so nothing is missing. But the thirteen diamonds are in both, and sixty-five over fifty-two is more than one. Third: a red card, and a king. Twenty-six and four.

7:09Thirty over fifty-two. That one fails both ways at once: the two red kings are counted twice, and twenty-four cards are never mentioned. Three groups, three verdicts: short, over, and neither. One more partition, and this one is deliberately lopsided. Throw two dice and add them. There are thirty-six ordered pairs, and the sum can be anything from two to twelve. Eleven possible sums, and every pair has exactly one of them, so the eleven sums are a partition of the thirty-six.

7:48Count the pairs behind each sum: one, two, three, four, five, six, five, four, three, two, one. Those eleven numbers add to thirty-six, which they have to, because every pair got counted exactly once. So the total of the eleven chances is thirty-six over thirty-six. One, as promised. And now look at the sizes. A sum of two happens one way and a sum of seven happens six ways. The two extremes are elementary events - one pair each, one and one, and six and six - while the middle one holds six pairs.

8:28Nine of the eleven are not elementary at all. Which kills the tempting shortcut. Eleven sums, so a chance of one eleventh each? No. Not one of the eleven is one eleventh, because one eleventh is not a whole number of pairs out of thirty-six. Eleven elevenths would total one perfectly happily. So the total coming to one can never be your evidence that the parts are equal. That last point deserves a number on it, because the temptation runs the other way too.

9:04If a group of events totals one, is it a partition? Take every partition of every experiment with up to seven outcomes. There are one thousand one hundred and fifty-five of them. Every single one totals exactly one. The conditions really do deliver. Now go the other way. Take every group of two or three events you can build on up to five outcomes - one thousand six hundred and sixteen groups - and sort them.

9:35The ones that cover everything without overlapping always total one. There are fifty-eight of those. The ones that cover everything but overlap always come out above one. Every one of the five hundred and eighty-six. The ones that do not overlap but leave something out always come out below one. All one hundred and ninety-two. And the ones that fail both conditions can land anywhere - and three hundred and nine of them land on one exactly.

10:09So three hundred and sixty-seven groups total one, and only fifty-eight of them are partitions. Fewer than one in six. Totalling one is what the conditions buy you. It is not evidence that you have them. So, the whole thing in one line. The probabilities of all the elementary events of an experiment add up to one. Not because of anything special about being small, but because those events cover every outcome and share none.

10:41Any other list of events with those two properties gives the same total - three colours out of nine marbles, two descriptions of a die, eleven sums of two dice. And any list without them will not, except by accident. If you are ever asked what the total comes to and the answer has been left blank, one is the only thing that fits. But knowing why is worth more than knowing the answer, because the why is what tells you when the answer does not apply.