Three conditions on a function, in place of a recipe for counting

0:00You already have a rule for probability. Count the outcomes that suit you, count all the outcomes, and divide. On a die, an even face is three faces out of six, so a half. That rule is about to be taken away from you, and replaced by something that looks weaker. So it is worth being precise about what it quietly assumes. It assumes the six faces are interchangeable. Take a die that has been weighted, so that the one comes up half the time and the other five faces share what is left equally.

0:37A half for the one, and a tenth each for the two, three, four, five and six. Now ask for an even face again. Two, four and six: a tenth, a tenth and a tenth. Three tenths, not three sixths. Nothing about that die is illegal. It is a perfectly good experiment. The counting rule simply does not describe it, and the counting rule has no way of saying so.

1:05What follows is a definition that does. The new definition starts by saying what kind of thing probability is. It is a function. So the first two questions to ask of it are what it takes in, and what it gives back. It gives back a single number, somewhere between nought and one, with both ends allowed. And what it takes in is not an outcome. It takes in an event, which is to say a collection of outcomes.

1:36That distinction is easy to skate over and it is the reason the whole thing works. The die has six outcomes. It has sixty-four events, because every collection you can make out of those six faces is one. Two outcomes carry four events, three carry eight, six carry sixty-four, and eight carry two hundred and fifty-six. The number of events doubles every time one outcome is added. So the function is being asked about far more things than there are faces on the die.

2:11The impossible event is in that collection, and so is the sure event. The face showing three is not in it. The event holding the face showing three is. Keep that separation and the three conditions coming next will read as conditions on a function. Lose it and they will read as arithmetic about faces. Here is the first condition. No event is ever assigned a negative number. That is all it says.

2:43It is easy to nod at and it is doing more than it looks. It rules out any construction, however natural it seemed on the way in, that could hand back a number below nought. And it is stated inclusively. Nought itself is permitted. That single word decides real cases later on, so it is worth holding on to now. An event may be assigned nought and still be a perfectly respectable event.

3:12The condition is a floor, not a demand for positivity. The second condition fixes the scale. The sure event is assigned the number one. Something is going to happen, and that something is worth one. This is a choice, and it is worth seeing it as one. Nothing in the mathematics required one. A hundred would have worked, and everything after this would have been percentages instead. One is chosen because it makes the arithmetic that follows come out in fractions rather than in hundredths.

3:50But notice what this condition does not say. It says nothing about any other event. It pins one value out of sixty-four on a die, and leaves the rest entirely open. The third condition is the one carrying the weight. Take two events that cannot both happen, meaning no outcome answers to both. The event of getting one or the other is assigned exactly the sum of the two numbers.

4:20Adding is permitted. That is the whole content. But read the restriction, because it is not decoration. The two events must not be able to happen together. If they can, the condition says nothing at all about their sum, and you may not use it. The reason is not mysterious. If an outcome sits in both events, then adding the two numbers counts that outcome twice, and the total is too big.

4:50Which is exactly the seam from the previous topic, arriving here as an arithmetic fault rather than a bookkeeping one. So the condition on pairs that cannot both happen is not a convenience. It is the precise circumstance under which adding is safe. And here is something the restriction hides, which is worth finding. Take four outcomes rather than six, to keep the counting small. Four outcomes give sixteen events, and those sixteen events make one hundred and twenty pairs.

5:27Eighty of those pairs overlap. Forty cannot both happen. Assign the four outcomes equal shares and check every pair. All forty of the excluding pairs add correctly, as the condition promises. And not one of the eighty overlapping pairs does. Every single one breaks. So on that assignment the restriction is exactly visible: adding works when the events exclude and fails when they do not. Now change the assignment. Give the first outcome the whole of it, one, and the other three nought each.

6:05All forty excluding pairs still add. Nothing has changed there. But of the eighty overlapping pairs, fifty-two now add correctly as well. Only twenty-eight break. And the twenty-eight are exactly the pairs where both members hold the outcome carrying the whole value. Nothing else can be double-counted, because everything else is worth nothing. So a student checking whether adding works, on that assignment, would find it working most of the time on pairs where it is not licensed.

6:41That is why the condition is stated on the events and never on the arithmetic. You cannot read off whether you were allowed to add by looking at whether the answer came out right. Three conditions, and now something follows from them. The impossible event is assigned nought. That was not one of the conditions. It is a consequence, and it needs only the third one. Here is the argument.

7:11Take any event, and pair it with the impossible event. Those two cannot both happen, because nothing at all answers to the impossible one. So the third condition applies, and the value of one or the other is the sum of the two values. But one or the other is just the original event again. Adding nothing to a collection changes nothing. So the event's value equals itself plus the impossible event's value.

7:42And the only number you can add to something without changing it is nought. Rather than asserting that, it can be searched for. Offer six candidate values for the impossible event, from minus one up to one, and for each one, hand it to the impossible event and then test the additive condition on every pair of events that cannot both happen. Exactly one candidate survives, and it is nought.

8:11The other five break at least one pair. Notice what that argument did not use. It never asked whether any value was negative, and it never used the sure event. Run it on a table of numbers that fails both of those conditions outright, and it still returns nought. The first two conditions are not needed here at all. Though something is needed: there has to be a real event for the argument to work against.

8:42Over an outcome list holding nothing at all, there are no pairs, nothing constrains the impossible event, and all six candidates survive. One outcome is already enough to pin it. The impossible event is worth nought. It is very tempting to read that backwards. If an event is worth nought, does that mean it is empty? That it cannot happen? It does not, and the difference matters. Take six outcomes, and this assignment: the first outcome gets one, and the other five get nought each.

9:20That satisfies every condition. Nothing is negative, the sure event is worth one, and adding works wherever it is licensed. Now count the events worth nothing under it. There are thirty-two of them, which is half of all sixty-four. Every event that leaves out the first outcome is worth nothing. And thirty-one of those thirty-two are not empty at all. The face showing two is worth nought, and the two is still on the list of outcomes.

9:52It has not been deleted from the experiment. It has been assigned nought. Compare that with the even assignment, where the only event worth nothing is the impossible one. So one direction is proved and the other is simply false. Impossible implies nought. Nought does not imply impossible. Now the useful consequence, and the reason all of this is workable at all. An event is a collection of outcomes. Each of those outcomes, on its own, is a one-outcome event.

10:29No two of them can both happen, because the outcomes are different. So the third condition applies to them, one at a time, all the way along. An event's value is the sum of the values of the outcomes it holds. There is no choice about it. Which means the whole function is determined by what it does to the one-outcome events. On a die, six numbers determine sixty-four values.

10:57On eight outcomes, eight numbers determine two hundred and fifty-six. And you can test that claim rather than believing it. Two assignments agreeing on all six one-outcome events agree on every one of the sixty-four events, with nothing left over. Now move one of those six values, and only that one. The two assignments no longer agree on the one-outcome events, and they differ at exactly one outcome. Thirty-two of the sixty-four events change value.

11:32Exactly half. And the half that changes is exactly the events holding the outcome you moved, which is what you would want. That is not a fact about six. On two outcomes it is two of the four, on three it is four of the eight, on four it is eight of the sixteen, on five it is sixteen of the thirty-two. Always exactly half, because an outcome is either in an event or out.

12:02So a candidate for a probability is now a small thing. One number against each outcome, and that is the whole of it. Take one coin. Two outcomes, so two numbers. A half and a half. Nothing is negative, and they total one. That is admissible. Now a quarter and three quarters. Nothing is negative, and they total one. That is admissible too. Both of those pass. Not one of them is more legal than the other.

12:34And that is not two answers, it is the beginning of an unlimited supply. Any number at all from nought to one, against one minus it. Sweep sixty-one values of it, evenly spaced from nought up to one, and every single one is admissible, with no exceptions, and no two of them the same assignment. Including the two ends. Nought against one is admissible: a coin that never comes up heads.

13:05That is not a contradiction, and it is not a defect in the conditions. The conditions were never supposed to tell you what the probability of a head is. They tell you which proposed answers are allowed to be called probabilities. Which one is right for a particular coin is a question about that coin, and it is not a question mathematics can settle. So testing a candidate comes down to two questions.

13:36Is every entry inside the permitted stretch, from nought to one? And do the entries total one? Those look like one question wearing two hats, and they are not. Sort candidates by both answers and there are four boxes, not two. Take three outcomes, and let every entry be a whole number of halves from minus one up to two. That gives three hundred and forty-three candidate rows. Six of them pass both tests.

14:07Twenty-one keep every entry inside the stretch and fail on the total. Thirty total one and carry an entry outside the stretch. And two hundred and eighty-six fail both. All four boxes are occupied, so neither test implies the other. Here is one from the third box, in plain numbers. Minus a half, one, and a half. Add them: minus a half plus one is a half, plus a half is one.

14:36It totals one exactly, and it carries a negative entry, so it is not admissible. And one from the second box: a half, a half and a half. Every entry is a number that could perfectly well be a probability. They total three halves. Also not admissible. Take the negative entries out of the grid, so nothing can go below nought, and one of the four boxes empties immediately. It is the box that needed a negative entry, and there is no other way to get into it.

15:13Now five candidates on six outcomes, tested one at a time. The first: one sixth against each face. Every entry is inside the stretch, and six sixths total one. It passes. The second: one against the first face, and nought against each of the other five. Nought is permitted and one is permitted, and the entries total one. It passes as well. The third: an eighth, two thirds, a third, a third, minus a quarter, minus a third.

15:47There are two negative entries, so it fails the range test at once. The fourth: a twelfth, a twelfth, a sixth, a sixth, a sixth, and three halves. Three halves is above one, so it fails the range test too. The fifth: nought point one, nought point two, nought point three, nought point four, nought point five, nought point six. Every one of those is a perfectly legitimate probability on its own.

16:15Add them and you get two point one. So it fails the total, and only the total. Two of the five pass. Three fail. But stopping at the first fault leaves the sharpest thing on the board unsaid. Go back to the third row and add it up. An eighth, two thirds, a third, a third, minus a quarter, minus a third. Over twenty-four: three, sixteen, eight, eight, minus six, minus eight.

16:45That is twenty-one twenty-fourths, which is seven eighths. Not one. So the third row fails the total as well. Now the fourth row. Over twelve: one, one, two, two, two, and eighteen. Twenty-six twelfths, which is thirteen sixths. Not one either. The fourth row fails the total as well. So two of the three failures fail both tests, not one. And exactly one row of the five is a clean single failure.

17:16It is the fifth, the row of decimals, and that is what makes it worth more than the other two. Every entry in it is a legitimate probability. It is rejected purely because six legitimate probabilities can total more than one. That single row is the whole argument that the two tests are independent. The other two rows, being wrong in both ways at once, could never have shown it.

17:46One more thing about the second row, the one with a single one and five noughts. It is common to describe the first condition loosely, as saying the entries should be positive and less than one. Read that way, the second row is rejected twice over. Its one is not less than one, and its noughts are not positive. Read as stated, with both ends allowed, it passes. So which reading you use decides a row.

18:16Run both readings over the same five candidates. The stated reading admits two of them, the first and the second. The strict reading admits only the first. They disagree on exactly one row of the five, and that row is the second. The condition as written allows both ends, so the second row is admissible, and the loose description is the thing that is wrong. This is not pedantry. An assignment giving one outcome the whole of it, and everything else nothing, describes an experiment whose result is already settled.

18:56That is an unusual experiment, not an illegal one, and the conditions were written to let it in. There is one more number floating around this definition that deserves a closer look. You will often see it said that a probability lies between nought and one. The lower end is one of the three conditions. The upper end is not. It does not have to be assumed, because it follows.

19:25Take any event, and take everything it leaves out. Those two cannot both happen, since no outcome is both in a collection and out of it. And between them they hold every outcome, so together they are the sure event. So by the third condition their two values add to the value of the sure event, which the second condition fixes at one. Now use the first condition. Neither of the two is negative.

19:54So neither can be more than one, because its partner would have to make up the difference by going below nought. Every event on the die passes that test: its value and its partner's always total exactly one. And both halves of the split are needed. Two events that cover the whole list while sharing an outcome do not split it, and neither do two that share nothing but leave something out.

20:22It has to be both, which is precisely the pair of conditions from the previous topic. So the upper end is not a fourth condition sneaked in. It is the first three conditions, applied to an event and its opposite. It is worth asking how much freedom these three conditions actually leave. On six outcomes the free data is six numbers, and they can be any non-negative numbers totalling one, so there are infinitely many admissible assignments.

20:56Restrict the entries to sixths, just to make the question answerable. How many tables of six sixths, none negative, totalling one? Four hundred and sixty-two. Loosen it to twelfths and the count goes up to six thousand one hundred and eighty-eight. That count can be reached two entirely separate ways, and it had better agree. One way tries every row on the grid and keeps the ones passing both tests.

21:27The other never builds a failing row at all: it hands out the units among the outcomes, so the total is one by construction. On small cases where both routes can be run, they agree exactly, which is the only reason the larger counts can be trusted. And the counting rule, the one you started with, is one of those four hundred and sixty-two. One entry in a very long list.

21:55That is the honest picture of what has happened here. The rule you had has not been contradicted. It has been demoted from a definition to a candidate. So what do the three conditions actually buy? They do not tell you the probability of anything. One coin has infinitely many admissible answers, and the conditions have no preference among them. What they do is tell you which proposed tables of numbers are allowed to be called probabilities at all.

22:29That is a test of candidates, not a computation of chances. And it is worth being clear about why that is an improvement rather than a retreat. The counting rule works only when the outcomes contribute equally. The conditions never mention the outcomes contributing equally, so they cover the weighted die, the bent coin, and every experiment where they do not. What was given up was the ability to compute an answer from nothing but a count.

23:01What was bought was a definition that survives the experiments where that count was never the right thing to do. And the counting rule is not gone. It comes back in the next topic, as a consequence rather than a definition, under a hypothesis that has to be stated out loud: that the outcomes are equally likely. Which was the assumption sitting silently inside it all along.