> If someone says they're "37% sure" tomorrow will rain, what does that mean exactly?
When someone says they're "37% sure" tomorrow will rain they mean that they assign the same probability to "tomorrow will rain" that they do to "if you throw three dice you'll get 9 or less" or "when you threw three dice you got 9 or less". In the second case the event is either true or false already and there is no uncertainty for you, their probability assignment is their best guess with the information they have.
The question is what is probability according to the subjective interpretation of probability? The answer usually given is that probability is a degree of belief. Thus, a probability of 37% means that you're 37% sure that some event will take place. What I'm saying is this definition is meaningless unless you define what it means to be X% sure about something, but the definition of "being X% sure" must not rely on the notion of probability because "probability" is what we are trying to define in the first place!
And what I try to explain is that one way to define what it means to be X% sure about A is to say that
- you put a number on it p(A)
- which is between 0 and 1
- and allows you to compare how sure you are about different things p(A) and p(B)
This number can be used to compute how sure you are about composite things:
p(A or B) = p(A) + p(B) - p(A and B)
p(A and B) = p(A given B) p(B) = p(B given A) p(A)
That number p happens to correspond to the notion of probability, but it has not been defined using a pre-existing notion of probability: https://en.wikipedia.org/wiki/Cox%27s_theorem
It's not completely arbitrary, it represents the degree of plausability you assign to the event.
These numbers have to obey some rules if you require that a set of beliefs is consistent.
The number you assign to the plausability of A and the number you assign to the plausability of not-A have to sum 1.
If you think A and B are equally plausible, you have to put the same number on them.
If you think that A and not-A are equally plausible, you have to assign the number 0.5 to both.
If you put the number p(head)=p(tails)=0.5 as your degree of plausability that the coin I just flipped (I actually did it!) is showing head or tails it's not an "arbitrary" number. It means that you think both (exhaustive) outcomes are equally plausible. Why do you say it cannot mean anything?