Counting Turns Into Chance
Last timePutting It Back, or Not
A probability is a count divided by a count. The division is easy; the work is making sure both halves count the same kind of thing and that the outcomes really are equally likely.
Four lessons of counting have produced numbers with no use yet. This lesson
gives them one, and it takes a single line to state.
- the thing described, such as the hand being a flush or the two dice adding to seven
- the length of the full list of possibilities, which is the count the previous lessons produce
- how many entries of that same list match the description. It must be counted from the same list, which is the trap of this lesson
- a number between nought and one. Nought means it never happens and one means it always does, and both of those come out of the formula rather than being added as rules
The assumption that is not written down
The formula divides by the total, which treats every outcome in the list as
carrying the same weight. Sometimes that is justified and sometimes it is
nonsense.
It is justified for a shuffled deck, a fair die, a fair coin, or a hash
function doing its job, and the justification is always symmetry: there is no
feature of the object that distinguishes one outcome from another, so they
cannot have different chances.
It is nonsense for most things observed in the world. The possible delays of
your train are a finite list, and counting them and dividing by their number
would tell you that an hour late is as likely as on time. The list being finite
is not the issue. The outcomes having equal weight is, and no amount of
careful counting repairs it.
So the question to ask before using the formula is whether anything about the
situation makes one outcome different from another. When the answer is yes, the
fraction is not a probability, and what is needed is measurement rather than
counting, which is where the course on samples picks the problem up.
Both halves, the same list
Here is the error that catches everybody once.
The lesson stops here
3 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents