You are one of the best long-range snipers in the World. You often have to eliminate targets standing one or two miles away and rarely miss. Crucially, this is not because you are better than others at aligning the scope of your rifle with the target’s head before taking the shot. Rather, this is because you are better at accounting for the external factors that will influence the bullet’s direction (e.g., the wind, gravity, etc.). Estimating these external factors is crucial for your success. One day, you are on a roof with binoculars, looking at a building four miles away that you and your team know is filled with terrorists and their innocent children. Your superior is beside you, looking in the same direction. Your mission is to inform your allies closer to the building if you see any movement. After a long wait, the terrorist leader comes out, holding a child close to him. He knows ennemies might be targetting him and would not want to hurt the kid. They are hastily heading towards another building nearby. You grab your radio to inform your allies but your superior stops you and hands you your sniper. “He’s so exposed. This is a golden opportunity for you.” she says. “I reckon we’ve got two minutes before they reach the other building. Do you think you can get him?“. You know that your superior generally accepts risking the lives of innocents but only if there are more bad guys than innocents killed, in expectation. “Absolutely not.” you respond. “We are four miles away! I’ve never taken a shot from this far. NO ONE has ever hit a target from this far. I’m just as likely to hit the kid.” Your superior takes a few seconds to think. “You always say that where the bullet ends up is the result of an equation: ‘where you aim + the external factors’, right?” she says. “Yes,” you reply nervously “And I have no idea how to account for the external factors from that far. There are so many different wind layers between us and the target. The Earth’s rotation and the spin drift of the bullet matter also a hell of a lot from this distance. I would also have to factor in the altitude, humidity, and temperature of the air. I don’t have the faintest idea what this sums up to overall from this far! I’m clueless. I wouldn’t be more likely to get him and not the kid if I were to shoot with my eyes closed.” Your superior turns her head towards you. “Principle of indifference,” she says. “You have no clue how external factors will affect your shot. You don’t know if they tell you to aim more a certain way rather than another. So simply don’t adjust your shot! Aim at this jerk and shoot as if there were no external factors! And you’ll be, in expectation, more likely to hit him than the kid.”
I'm curious whether people agree with the superior. Do you think applying the POI is warranted, here? Any rationale behind why (not)?
EDIT: You can assume the child and the target cover exactly as much hittable surface area if you want, but I'm actually just interested in whether you find the POI argument valid, not in what we think the right strategic call would be if that was a real-life situation. (EDIT 2:) This is just a thought experiment aiming at assessing whether applying POI makes sense in situations of complex cluelessness. No kid is going to get killed in the real world because of your response. :)
I think the principle is fine when applied to how variables effect movement of the bullet in space. I don't necessarily think it means taking the shot is the right call, tactically.
Note: I've never fired a real gun in any context, so a lot of the specifics of my reasoning are probably wrong but here goes anyway.
Essentially I see the POI as stating that the bullet takes a 2D random walk with unknown step sizes (though possibly with a known distribution of sizes) on its way to the target. As distance increase, variance in the random walk increases.
Given typical bullet speeds we're talking about >6 seconds to reach the target, possibly much more depending on drag. And the sniper is actually pointing the rifle so it follows a parabolic arc to the target. In that time the bullet falls .5*g*t^2 meters, so the sniper is actually pointing at a spot at least 180m above his head, possibly 500m if drag makes the expected flight speed even a few seconds longer. More still to the extent the distance means the angle of the shot has to be so high you need to account for more vertical and less horizontal velocity plus more drag. Triple the distance means a lot more than 3x the number of opportunities for random factors to throw off the shot. The random factors are playing plinko with your bullet.
After the first mile (limit of known skill) the expected distance of where the bullet lands from the target increases. At some sufficiently far distance, it is essentially landing in a random spot in a normal distribution around the intended target, and whether it hits the terrorist or the child is mostly a function of how much area each takes up (the adult is larger), and the extent to which one is blocking the other (not stated). Regardless, when the variance is high enough, the most likely outcome is "neither." It hits the ground meters away, and now the terrorists all know you took the shot and from what direction. If the terrorist just started walking, there might be a better chance of hitting the building than anything else.
So, in a strict probabilistic sense, yes, your probability of hitting the terrorist is still higher than hitting the child. If that is the superior's sole criterion for decision making, they've reasoned correctly. That is not the sniper's decision making threshold (he wants a higher certainty of avoiding the child). I would expect it is also not the higher-ups' sole criterion, since it is most likely going to fail and alert the enemy about their position as well as the limits of their sniping capabilities. I have no idea whether the sniper has any legally-defensible way to refuse to follow the order, but if he carries it out, I don't think issuing the order will reflect well on the superior.
That said: In practice a lot would depend on why the heck they stationed a sniper three miles away from a target with no practical way to turn that positioning into achieving this goal. The first thought I have is that either you're not where you're supposed to be, or the people that ordered you there are idiots, and in either case your superior is flailing trying to salvage the situation. The second is that you're not really expected to take out this target at this range, and your superior is either trying to show off to his superior, or misunderstood his orders, or wasn't told the real reason for the orders. The third is that of course the sniper would have brought this up hours ago and already figured out the correct decision tree, it's crazy this conversation is only happening after the terrorist leaves the building.
I mentioned this in my comment above, but I think it might be worthwhile to differentiate more explicitly between probability distributions and probability density functions. You can have a monotonically-decreasing probability density function F(r) (aka the probability of being in some range is the integral of F(r) over that range, integral over all r values is normalized to 1) and have the expected value of r be as large as you want. That's because the expected value is the integral of r*F(r), not the value or integral of F(r).
I believe the expected value... (read more)