I'm not sure there is a difference between active disbelief, or even militant disbelief, and simple lack of belief. In other words, active disbelief is not belief in nonexistence. Rather, it is vociferous withholding of belief. It has been so long since I read Dawkins that I cannot remember his precise position on this. Speaking for myself, my active, at times militant disbelief is in celebration of the human logical faculty and freedom to choose rationality over superstition.

@Flyingsaucesir
But we know that buses exist from our experience. In that case you can also assign a probability if you know how many buses there are and how frequent they drive by. That is not the case with the god-claim. You have no prior experience either way that justifies a probability.

@Dietl
We have the human propensity for inventing gods, the biological basis for the religious experience, the suffering of innocents, AND a complete lack of independently-verifiable evidence of existence. I think this is sufficient information for the individual to make a determination of probability. If that turns out to be one in ten or one in a billion, that's alright. As long as the probability is not zero, one is on solid ground. One does not take on any special responsibility for proving anything. It is simply a judgement call, the kind we make every day when we cross the street.

@Dietl
By the way, about special pleading: we only have one proposition before us at the moment. If we render a judgement on the likelihood of that one thing, we are not automatically responsible for rendering judgements on something else. We can take everything on a case-by-case basis. For instance, if you want to talk extraterrestrial species, then we can roll out the Drake equation and roughly calculate their existence without considering a host of other things.

Regarding your first comment. It seems to me that you are arguing that the probability of a god existing or not existing depends on what you and I think or evolved to think. Like if you flipped a coin and evolved to prefer heads, would you say that somehow influences the coinflip? I hope this seems as ridiculous to you as it does to me. Please tell me if I'm mischaracterizing your comment.
About my argument using special pleading: What I wanted to point out was that if you believe one thing without evidence you have no justification not to also believe in another thing.It's just one argument that you can ignore if you want, that's not unreasonable. But there is a difference between aliens and gods. The alien claim is falsifiable. We can make predictions about what we would have to find in case aliens exist on a planet, which we can test. Also we know about one instance of life evolving on a planet: Earth. So we are safe to assume that life potentialy evolved elsewhere. We have nothing of that sort for the god claim.