Favourable over total: the definition this chapter runs on

0:00Two ideas are already on the table, and this is where they pay off. The first: sometimes the outcomes of an experiment are interchangeable, so that nothing about the set-up prefers one to another. The second: a definition that needs no trials is worth having, because most questions worth asking cannot be rehearsed. Put those together and you get a rule you can compute with, and it is the rule the whole subject runs on.

0:32It is short enough to memorise in ten seconds, which is exactly the problem. Memorised without its reason, it produces wrong answers that look right. So this is the rule, and then the reason, and then the one distinction that decides almost every mistake anybody makes with it. Here it is. The probability of an event is the number of outcomes that would make it happen, divided by the number of outcomes there are altogether.

1:06Favourable over total. Two counts, and both of them count the same kind of thing: outcomes. Not descriptions. Not answers. Not the numbers the question happens to mention. Outcomes. And there is a clause attached to it that is part of the rule and not a footnote to it. All of this holds provided the outcomes are equally likely. Keep that clause on the board, because everything else in this video is either the reason for it or the price of forgetting it.

1:41Now the reason, because a ratio handed over with no reason is exactly the kind of thing that gets misapplied. Suppose an experiment has six outcomes and nothing separates them. Then each one has to carry the same share of the certainty, and the shares have to fill the experiment exactly, because the experiment always ends somewhere. Six equal shares filling one whole. So each is a sixth. Draw it as a strip cut into six equal cells, each labelled one sixth.

2:17Now an event is not a new thing. It is just some of those cells. Shade the cells it covers and add them up. Two cells shaded, each a sixth, so a sixth plus a sixth: two sixths. And that is what favourable over total is. It is not a definition of chance handed down from nowhere. It is what you get when equal weights are added. Which is why the equal-likelihood clause is doing the work: without it the cells are different sizes, and adding two of them tells you nothing about how many there were.

2:56Watch what the clause is holding up. Grey out the clause, keep the ratio, and the ratio still computes. It gives you a number every time. That is the danger. A bag with four red balls and one blue one has two colours, and one over two is not the chance of red. The chance of red is four fifths, because the balls are the interchangeable things and the colours are not.

3:24So the rule never applies to an experiment. It applies to a list of outcomes you have chosen, and it is legitimate only when nothing separates the things on that list. Which means the first question is never what is the probability. The first question is: what is my list, and is anything on it different from anything else on it? Here is the distinction that decides almost every wrong answer in this subject.

3:56Outcomes are what the experiment can actually end in. Descriptions are the words you put around them. One description can cover several outcomes. Several descriptions can cover one. The rule counts outcomes. It has nothing at all to say about how many descriptions you wrote down. The most common error is the same sentence every time: there are two ways this can go, so it is a half. Two ways this can go is a statement about descriptions.

4:29A second error rides in beside it: favourable over unfavourable. Anybody who has met odds imports this without noticing, and it is a different quantity with a different denominator. Favourable over everything. Both counts, over the same list. So, the simplest case, done slowly. One coin, tossed once. List the outcomes: head, tail. Two of them, and the two faces of a fair coin are interchangeable. The event is: it comes up heads.

5:05How many outcomes make that happen? One of the two. So the probability of a head is one half. Now the tail. And here is the thing worth noticing. You do not get one half for the tail by subtracting the head's answer from one. You get it by counting again. One outcome out of two makes a tail. One half. The two happen to add to one, and that is a fact you can check rather than a rule you applied.

5:34Nothing was tossed. No coin moved. The whole thing was settled by listing and counting. Now three, so the pattern shows. A bag holds one red ball, one blue ball and one yellow ball, all the same size, and you draw one without looking. That phrase, all the same size, is not scene setting. It is the guarantee that your hand cannot prefer one of them. Delete it and the computation is not justified.

6:07Three outcomes. Red, blue, yellow. The chance of yellow: one outcome out of three, so a third. The chance of red: one out of three, a third. The chance of blue: one out of three, a third. Three separate questions, three separate numerators, and one denominator that never moved. And the three thirds fill the experiment exactly, which is the weight strip again, seen from the other side. Now an event made of more than one outcome, which is where the counting starts to earn its keep.

6:45A die is thrown once. Six outcomes: one, two, three, four, five, six. The event is: the result is greater than four. Do not divide yet. Turn the description into a list. Which of the six results are greater than four? Five, and six. That is the whole step, and it is the step people skip. Two outcomes favour it, out of six, so two sixths, which is a third.

7:13Now compare that with what the two wrong readings would have said. Two ways this can go, above four or not above four, so a half. Wrong. Two favourable against four unfavourable, so two over four, a half again. Also wrong, and wrong by the same amount. The true answer is a third and both wrong readings say a half. They are out by a sixth, on a quantity that is only a third to begin with.

7:44Now ask the same die for the rest. The event is: the result is four or less. Turn it into a list first. One, two, three and four. Four outcomes favour it. And out of how many? Six. The same six as before. Four sixths, which is two thirds. Put the two answers side by side and look at what moved. The numerator went from two to four. The denominator did not move at all.

8:13It could not have. The denominator is the size of the outcome list, and the outcome list belongs to the experiment, not to the question. One die, one list of six, and as many questions as you like sitting on top of it. And a third plus two thirds is one, which is the strip filled: every cell is in one of the two events and none is in both.

8:41So the whole method is four steps, and only one of them is arithmetic. Write down every outcome the experiment can end in. Check that nothing separates them, and if something does, go to a finer list until nothing does. Cross off the outcomes the description does not cover, and count what is left. Then divide, by the length of the whole list. The reason to insist on the list is that a description can hide its own size.

9:12Greater than four sounds like one thing and is two. Four or less sounds like one thing and is four. Written as words they look like a matched pair. Written as lists they are two against four, and the answer is obvious before any dividing happens. It is worth knowing how often the wrong readings happen to be right, because that is why they survive. Take every experiment with up to eight outcomes and every event you can build inside one. There are five hundred and ten of them.

9:49Favourable over unfavourable gets the right answer eight times. Once per experiment, and always in the same silly case: when no outcome at all is favourable and both readings are nought. Everywhere else it is wrong. That reading is not a shortcut, it is a different quantity. Two ways this can go, so a half, does better: it is right ninety-eight times out of five hundred and ten. Under one in five. Often enough to feel like a rule, rarely enough to be one.

10:24And the thing the whole rule rests on holds everywhere it should. Take any two events with no outcome in common: the chance of one or the other is the chance of one plus the chance of the other, on all one thousand and ninety-two such pairs. Take two that do share an outcome and it fails on every single one of the four thousand three hundred and sixty-eight of them, because the shared cell got added twice.

10:55That is the strip talking. Weights add when the cells are separate and not when they are shared. One last thing, because this rule has a history and it is a short one. The subject starts in the sixteenth century with Cardan, an Italian doctor who wrote the first treatise on the mathematics of gambling games. Then James Bernoulli, sixteen fifty-four to seventeen oh five. Fifty-one years. Abraham de Moivre, sixteen sixty-seven to seventeen fifty-four. Eighty-seven years.

11:32And Pierre Simon Laplace, seventeen forty-nine to eighteen twenty-seven, who wrote the definition you have just been using in seventeen ninety-five, and his great work on the subject seventeen years later. More than two centuries on, it is used in biology, economics, genetics, physics and sociology. And it is still the same two counts, over the same list, with the same clause attached. Favourable over everything, once you have earned the right to say the outcomes are interchangeable.

12:09The counting is the easy half. The list is the hard half, and it is the half that decides the answer.