The interesting question isn't so much "how do I convert a degree of belief into a number", but "how do I reconcile my degrees of beliefs in various propositions so that they are more consistent and make me less vulnerable to Dutch books". One way to do that is to formalize what you take that statement to mean, so that its relationships to "other beliefs" becomes clearer. It's what, in the example you suggest, the Doomsday clock scientists have done. So you can look at whatever data has been used by the Doomsday Clock people, and if you have reason to believe they got the data wrong (say, about international agreements), then your estimate would have to be different from theirs. Or you could figure out they forgot to include some evidence that is relevant (say, about peak uranium), or that they included evidence you disagree is relevant. In each of these cases Bayes' theorem would probably tell you at the very least in what direction you should update your degree of belief, if not the exact amount. Or, finally, you could disagree with them about the structural relationships between bits of evidence. That case pretty much amounts to making up your own causal model of the situation. As other commenters have noted it's fantastically hard to apply Bayes rigorously to even a moderately sophisticated causal model, especially one that involves such an intricately interconnected system as human society. But you can always simplify, and end up with something you know is strictly wrong, but has enough correspondence with reality to be less wrong than a more naive model. In practice, it's worth noting that only very seldom does science tackle a statement like this one head-on; as a reductionist approach science generally tries to explicate causal relationships in much smaller portions of the whole situation, treating each such portion as a "black box" module, and hoping that once this module's workings are formalized it can be plugged back into a more general model without

The problem with your question is that the event you described has never happened. Normally you would take a dataset and count the number of times an event occurs vs. the number of times it does not occur, and that gives you the probability. So to get estimates here you need to be creative with the definition of events. You could count the number of times a global war started in a decade. Going back to say 1800 and counting the two world wars and the Napoleonic wars, that would give about 3/21. If you wanted to make yourself feel safe, you could count the number of nukes used compared to the number that have been built. You could count the number of people killed due to particular historical events, and fit a power law to the distribution. But nothing is going to give you the exact answer. Probability is exact, but statistics (the inverse problem of probability) decidedly isn't.