Yes, I think it goes in our background knowledge that E was a result of two coin tosses, so ~E is a different result of two coin tosses. But what if E isn't the result of a test?
H = Bob is going to die
E = Bob is playing in the street.
This example isn't qualified enough for analysis. E isn't quite a premise--it isn't either true or false. It's sometimes true, and sometimes false. If E is, for damned sure, true, it could be false in ten minutes. That needs to be qualified more.
Same with H. Technically, H at large can be considered an event by proxy--is Bob going to die, at all? But in this sense, P(H)=1. Sure. Bob's a mortal--poor mortals.
What happens in practice with these particular premises, since the intent is presumably to figure out if Bob's going to die because he's out in the street, is that P(E) is going to depend on how long he's playing in the street.
Again, E isn't even a premise--P(E) doesn't even have a fixed value (it changes while Bob's playing, and depends on how long he does). So it simply can't be analyzed from a Bayesian approach. Fortunately this can be solved, in this particular case, by qualifying E some more.
It depends. Knowing the outcome of a single coin toss is little better than no coin toss at all.
I'm talking about a different sort of thing. In the coin toss case, you know there will be two options. Only a bad analogy can be made to this case--it would require considering the possibility that something you've never observed happened... something like, the coin turns up fingers.
You don't even know if they make coins with fingers on them. You know there are coins with heads, and coins with tails, and two headed coins even. Coins with fingers are entirely different. Once you know, for sure, that someone, somewhere, even makes a coin with fingers in the first place, something very significant happens.
This is about instances of classificiations existing, depending on whether or not anything else of that class is known to exist at all.
Edit: actually a single-coin toss provides some confirming evidence.
Correct. It does. But it does for a different reason than the thing I'm talking about.
Knowing that life occurred on Earth doesn't help us with respect to ET life because we don't know if there is something unique (or a combination of unique events) about Earth that isn't found on any other planet in the universe.
Yes and no. We don't know if it's unique. We can't get a probability out of a single data point. So it doesn't help us figure out what the odds are.
But from an epistemic standpoint, it is
tremendously more valuable than not knowing there is life (however that can happen), because it puts the foot in the door that there's life in the first place. There's some sort of condition--we don't know whether or not it's unique, but we know there is one--that leads to life. That's a huge step.