Perhaps not self evident, but easily provable. Now, having to COME UP WITH this when all you know is p(a) = x; p(b) = 1 - x, and the fact that p(a)p(b) == p(b)p(a) is some awesome insight.
You don't even need the first part. As long as p(a) and p(b) are both constant, they can add to any number, not just 1. They could even be negative, which is impossible for a physical coin of course but my point is: you only need to know that there are two constants, and multiplication is commutative, and you're done :)
An example: if you have a die and you don't know exactly how many sides it has, or whether it's fair, you can still get a perfectly fair 50/50 outcome. Just count "1 followed by 2" vs. "2 followed by 1" and ignore every other result.
Quantum mechanics: frustrating formal mathematicians for over a century.
You'd have to exercise caution when using (quasi-)probabilities less than zero or greater than one, because many of the classical properties of probability theory depend upon those conditions.
