Not good enough for Dean, a professor in the Wharton statistics department, who said "But you aren't willing to give a number? Good Bayesians / Game Theorists have to bet on everything!"
A good Bayesian puts a probability p on every future event E where one would be indifferent to taking a small bet that pays off 1-p if the event is true and loses p if the event is false.
As a computational complexity theorist we tend not to be Bayesians, rather look at worst-case performance under that assumption that everyone is working against us. But I've been talking with economists a bit recently so lets take the Bayesian approach.
Richard Lipton asked if one would bet their life that P≠NP. In some sense I have since the vast majority of my research becomes trivial or uninteresting if P=NP. But how much one bets isn't really the right question since that involves taking risk into account as well as beliefs.
So what odds do I give? The problem is that I could only bet conditional on P v. NP being solved in some reasonable amount of time which would alter my beliefs since a proof that P=NP requires only an algorithm but a proof that P≠NP requires showing no algorithm can work. And the no trade theorem comes into play: If someone were to offer me a bet on P v. NP, I'd secretly suspect that they knew an algorithm or a proof I hadn't seen yet. But suppose that I could make a bet against a magical oracle would reveal the correct answer once a bet is made.
Still I can't in my heart give a positive probability to P=NP. So the probability of P=NP is zero, but in the measure theory sense that because an event has probability zero it doesn't mean it can't happen, only that it won't.