There are several complications in the example you give, and I’m not sure which are intentional.
Let’s start with a simpler example. You somehow end up needing to take a 400 meter shot down a tunnel with an experimental rifle/ammo you’ve been working on. You know the rifle/ammo inside and out due to your work and there is no wind, but the rifle/ammo combination has very high normal dispersion, and all that is exposed is a headshot.
In this case, where you center your probability distribution depends on the value of the kids life. If the terrorist is about to nuke the whole earth, you center it on the bad guy and ignore the kid. If the terrorist will at most kill that one kid if you don’t kill him now, then you maximize expected value by biasing your distribution so that hitting the kid requires you to be further down the tail, and the ratio of terrorist/hostage hit goes up as the chance of a hit goes down. If the kid certainly dies if you miss, also dies if you hit him, and is only spared if you hit the terrorist, then you’re back to ignoring the kid and centering the distribution on the terrorist—even if you’re more likely to hit the kid than the terrorist.
In the scenario you describe, you don’t actually have the situation so well characterized. You’d be forced to lob bullets at a twenty degree inclination, without being able to use sights or see your target—among many other large uncertainties. In that case it’s not that you have a well known distribution and unknown result of the next draw. You don’t know what the distribution is. You don’t know what the meta distribution the distribution is being drawn from.
Statements like “more likely” are all relative to a model which you presuppose has some validity in context. What’s the model, and where do you think you’re getting the validity to say it? Even if the simulation God paused the game and spoke to you saying “I’ll run the experiment a billion times, and we’ll see if the kid gets hit more often”, you’d have no idea how to set up that experiment because you don’t know what you’re controlling for.
I’d guess that you’re asking about something in between, but I’m not sure which unknowns are intentional.
There are several complications in the example you give, and I’m not sure which are intentional.
Let’s start with a simpler example. You somehow end up needing to take a 400 meter shot down a tunnel with an experimental rifle/ammo you’ve been working on. You know the rifle/ammo inside and out due to your work and there is no wind, but the rifle/ammo combination has very high normal dispersion, and all that is exposed is a headshot.
In this case, where you center your probability distribution depends on the value of the kids life. If the terrorist is about to nuke the whole earth, you center it on the bad guy and ignore the kid. If the terrorist will at most kill that one kid if you don’t kill him now, then you maximize expected value by biasing your distribution so that hitting the kid requires you to be further down the tail, and the ratio of terrorist/hostage hit goes up as the chance of a hit goes down. If the kid certainly dies if you miss, also dies if you hit him, and is only spared if you hit the terrorist, then you’re back to ignoring the kid and centering the distribution on the terrorist—even if you’re more likely to hit the kid than the terrorist.
In the scenario you describe, you don’t actually have the situation so well characterized. You’d be forced to lob bullets at a twenty degree inclination, without being able to use sights or see your target—among many other large uncertainties. In that case it’s not that you have a well known distribution and unknown result of the next draw. You don’t know what the distribution is. You don’t know what the meta distribution the distribution is being drawn from.
Statements like “more likely” are all relative to a model which you presuppose has some validity in context. What’s the model, and where do you think you’re getting the validity to say it? Even if the simulation God paused the game and spoke to you saying “I’ll run the experiment a billion times, and we’ll see if the kid gets hit more often”, you’d have no idea how to set up that experiment because you don’t know what you’re controlling for.
I’d guess that you’re asking about something in between, but I’m not sure which unknowns are intentional.