Equally likely outcomes, and the everyday cases where that fails

0:00A coin goes up, and everybody in the room already knows the answer. One half. Two results, so one half each. That answer is right, and the reason almost everybody gives for it is wrong. Because here is a second experiment with two named results. A bag holding four red balls and one blue one, and you reach in without looking. Red, or blue. Two results. And the chance of red is four fifths, not one half.

0:31So counting the results you have named tells you nothing on its own. Something else has to be true before you are allowed to divide, and this video is about what that something is. Start with the coin, and ask what makes it a fair one. Turn it over. There is a head on one side and a tail on the other, and apart from the picture stamped on them the two faces are the same disc of the same metal.

1:00Nothing about the object prefers one to the other. That is the whole argument, and notice what kind of argument it is. It is a claim about the coin. It is not arithmetic, and no amount of arithmetic could have produced it. You had to look at the physical thing and decide that its two sides are interchangeable. A coin with lead poured into one face would still have exactly two results, and the argument would simply fail on it.

1:32So fairness lives in the object, before any fraction is written down. There is a second condition, and it is easy to miss because it usually travels quietly beside the first. The throw itself must not steer the result. A symmetric coin, released by a machine calibrated to flip it exactly two and a half turns onto a soft mat, comes down the same way every single time. The coin has not changed at all. Its two faces are still interchangeable.

2:08What has changed is that the toss is doing work. So there are two separate assumptions here, and it is worth pulling them apart. The coin is symmetric. And the toss is free. Both are claims about the world, imported from outside the mathematics, and neither of them is a theorem. One more thing gets quietly set aside, and it deserves to be said out loud rather than smuggled. A coin can land on its rim.

2:40On a hard floor that is vanishingly rare, and we ignore it. Drop the same coin into deep sand, though, and it will happily stand on its edge. So the two-result model is not a fact about coins. It is a decision about which surface we are thinking of. That is not a weakness. Choosing what to ignore is most of what modelling is. The honest version is: on a hard floor, a fairly made coin, freely tossed, has two results, and nothing separates them.

3:14Every clause in that sentence is doing work. A die carries the same story six times over. One, two, three, four, five, six. Six results, and a cube with nothing to choose between its faces. So each face gets one sixth. But be clear about what has happened here. Nobody proved that a die is fair. We declared it. From here on, when a die is thrown, we agree to treat the six faces as interchangeable - an assumption imported, not a result reached.

3:51You could test it, with enough throws, and the test would be an experiment rather than a proof. Knowing which of your statements are assumptions is not pedantry. It is the difference between using a model and believing one. Now back to the bag, because it is the example that breaks the pattern open. Four red balls, one blue ball, and a draw made without looking. Ask about the balls, one at a time, and the symmetry argument works perfectly.

4:24The five balls are the same size and the same weight, your hand cannot tell them apart, so no ball is preferred. Five results, one fifth each. Now ask about the colours instead. There are four ways to draw a red ball and one way to draw a blue one, so red comes out at four fifths and blue at one fifth. Four times as likely. The two colours are nothing like even.

4:52And nothing was changed in between. Same bag, same draw, same hand. Which means the assumption we have been making does not attach to the experiment at all. It attaches to the way you chose to carve the experiment into results. Carve that single draw into five balls and the results are even. Carve the very same draw into two colours and they are not. Same experiment. Two carvings. One of them earns you the right to divide and the other does not.

5:27So the question is never is this experiment fair. The question is always: are these results, as I have named them, interchangeable? And notice the repair when they are not, because students often think an uneven list means nothing can be computed. It means the opposite. Go back to a finer grain. The colours were uneven, but the balls underneath them were not, and counting the balls is exactly how we got four fifths.

6:00An uneven naming is fixed by naming more carefully, never by giving up. Here is the same error with the disguise taken off. Two coins are tossed together. Somebody says: the result is two heads, or two tails, or one of each. Three results, so a third each. Write out what can actually happen, and be careful to keep the two coins apart. Head and head. Head then tail. Tail then head. Tail and tail.

6:32Four results, and by the symmetry argument those four really are interchangeable. Two heads is one of the four, so a quarter. Two tails is one of the four, so a quarter. And one of each is two of the four, so a half. A quarter, a quarter, a half. Not a third, a third, a third. The mixed result was undercounted by one sixth, and the other two were each overstated by a twelfth.

7:00The three misses add up to nothing, which is exactly why the wrong answer looks so tidy. Now the same bad reasoning on a different question, and watch what happens. One die is thrown. It lands odd, or it lands even. Two results, so one half each. That answer is correct. Odd is one, three and five. Even is two, four and six. Three faces against three faces, so it really is a half.

7:33But the reasoning that got there was the same reasoning that failed on the two coins. It happened to land on the right number because the counts happened to be equal. Change the question to: does it land on a six, or not on a six. Two results again, and the same argument again offers a half. The truth is one sixth. That argument is out by a third, which on a quantity smaller than one is not a small error at all.

8:03An argument that is sometimes right is not a method. It is a coin toss of its own. So how often does a naming happen to be even? Take the six faces of a die and count every way of sorting them into two named results. There are thirty-one such ways. Ten of them are even. Twenty-one are not. So name two results on a die and reach for a half, and you are right about a third of the time.

8:34It gets sharper. Sort the five balls into two named results, and there are fifteen ways to do it. Not one of them is even, and the reason has nothing to do with colour. Five will not split into two equal parts. Whatever you paint on those balls, two names can never be even. The same trap catches the two coins. Four outcomes cannot be shared equally among three names, so no three-way description of two coin tosses is ever even. Not one.

9:09Widen it right out. Take any experiment with at most seven outcomes and list every possible way of grouping them into named results. There are one thousand one hundred and fifty-five. Forty-one are even. Under four in a hundred. Evenness is not the normal case that occasionally fails. It is the rare case you have to earn. Which is why the everyday examples are worth arguing about rather than ticking.

9:41A driver turns the key. The car starts, or it does not. Not even, because that depends on the state of the car, and most cars start. A player shoots at a basket. She scores, or she misses. Not even either, because that depends on her skill, and skill is exactly a reason to prefer one result. A true or false question is answered rightly or wrongly. This one is arguable, and the argument is the answer.

10:14If the person knows the material, the two are not even. If the person is guessing blind, nothing separates them, and they are. A baby is born a boy or a girl, and to a good approximation those are even. Notice that in every case the verdict came from thinking about the situation, never from counting the words in the sentence. And it is why a coin decides who kicks off.

10:42Not because a coin is magic, but because its two results are interchangeable before anybody has chosen a side, so no team can be favoured by the procedure. The fairness of the game start is the symmetry of the object, borrowed. So here is the whole idea in one line. Favourable over total is not a definition of chance. It is a shortcut that becomes available once you have argued that your results are interchangeable.

11:16The argument comes first, and it is about the world: a symmetric object, a free throw, a naming fine enough that nothing separates one result from another. Skip it and you get the bag, where two colours look like two coins and are four to one, or the two tosses, where three sensible-sounding descriptions hide four outcomes. Naming two results costs you nothing, so it cannot buy you anything either.

11:46Ask instead whether anything separates them. If nothing does, divide. If something does, look closer until you find a grain where nothing does. That is the entitlement. Everything else in probability is built on top of it.