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 of r in the stated scenario is large enough that missing is the most likely outcome by far. I am seeing some people argue that the expected distribution is F(r,θ) in a way that is non-uniform in θ, which seems plausible. But I haven't yet seen anyone give an argument for the claim that the aimed-at point is not the peak of the probability density function, or that we have access to information that allows us to conclude that integrating the density function over the larger-and-aimed-at target region will not give us a higher value than integrating over the smaller-and-not-aimed-at child region
So, as you noted in another comment, this depends on your understanding of the nature of the types of errors individual perturbations are likely to induce. I was automatically guessing many small random perturbations that could be approximated by a random walk, under the assumption that any systematic errors are the kind of thing the sniper could at least mostly adjust for even at extreme range. Which I could be easily convinced is completely false in ways I have no ability to concretely anticipate.
That said, whatever assumptions I make about the kinds of errors at play, I am implicitly mapping out some guessed-at probability density function. I can be convinced it skews left or right, down or up. I can be convinced, and already was, that it falls off at a rate such that if I define it in polar coordinates and integrate over theta that the most likely distance-from-targeted-point is some finite nonzero value. (This kind of reasoning comes up sometimes in statistical mechanics, since systems are often not actually at a/the maxentropy state, but instead within some expected phase-space distance of maxentropy, determined by how quickly density of states changes).
But to convince me that the peak of the probability density function is somewhere other than the origin (the intended target), I think I'd have to be given some specific information about the types of error present that the sniper does not have in the scenario, or which the sniper knows but is somehow still unable to adjust for. Lacking such information, then for decision making purposes, other than "You're almost certainly going to miss" (which I agree with!), it does seem to me that if anyone gets hit, the intended target who also has larger cross-sectional area seems at least a tiny bit more likely.
I used to use a ricer, but found that it always made the potatoes too cold by the time I ate them. Do you find this? If not, do you (even if you never thought of it this way) do anything specific to prevent it? If so, do you then reheat them, and how?
With a stand mixer and the whisk attachment I found removing the ricer step hasn't really mattered, but any other whipping method and yeah, it's very useful.
Fair enough, I moved into a small space a few years ago and mostly buy smaller quantities now. I also like that the Little Potato Company's potatoes are already washed and I'm often boondocking/on a limited water supply.
Costco is generally above average in most things, so definitely a good choice. I find the brands I mentioned to be more consistently high quality across locations and over time, but not too much better at their respective bests. So when I need a specific meal to be high quality, like on holidays, I'll make sure to go to Trader Joe's.
FWIW the Trader Joe's organic golds are around $4/3lb bag. The Little Potato Company's bags are around $2-3/lb. I have bought both in at least 10 states each at this point and those price have been fairly consistent. I also don't want to spend a huge amount on potatoes.
In my experience that's true for a hand-held masher or hand mixer, but if I'm slow-whipping in a stand mixer with butter and cream, golds give a fluffier, smoother, lighter result.
I really enjoyed this piece, not because of the specific result, but because of the style of reasoning it represents. How much advantage, under what kind of rules, can be overcome with what level of intelligence?
Sometimes the answer is none. "I play x" overwhelms any level of intelligence at tic tac toe.
In larger and more open games the advantage of intelligence increases, because you can do more by being better at exploring the space of possible moves.
"Real life" is plausibly the largest and most open game, where the advantage of intelligence is maximized.
So, exploring the kind of question the OP posits can give us a kind of lower bound on how much advantage humans would need to defeat an arbitrarily smart opponent. And extending it to larger contexts can refine that bound.
By the time we hit chess-complexity, against an opponent not trained for odds games, we're already at around two bishops odds for an uncommon-but-not-extreme level of human skill.
I think a lot of the problems that arise in discussing AI safety are a (in the best cases much more well reasoned) form of "You think an AI could overcome X odds? No way!" "Yes way!"
The latter. And yes, I do agree with the superior on that specific, narrow mathematical question. If I am trying to run with the spirit and letter of the dilemma as presented, then I will bite that bullet (sorry, I couldn't resist).
In real world situations, at the point where you somehow find yourself in such a position, the correct solution is probably "call in air support and bomb them instead, or find a way to fire many bullets at once- you've already decided you're willing to kill a child for a chance to to take out the target."
Similarly, if the terrorist were an unfriendly ASI and the child was the entire population of my home country, and there was knowably no one else in position to take any shot at all, I'd (hope I'm the kind of person who would) take the shot. A coin flip is better than certainty of death, even if it were biased against you quite heavily.
Interesting, why don't you like them for mashing? That's specifically what I like them best for. Although IIUC a knish needs a different texture to hold together well. I also don't use golds for (unbreaded) potato cakes unless I mash them in advance and use them left over.
I'm no chef, but I love to cook, and my thanksgiving meals are planned in spreadsheets with 10 minute increments of what goes where. Plus I currently live full-time in an RV so I've gotten used to improvising with nonstandard and less reliable tools. Take or leave my suggestions accordingly.
It's often a good idea, until and unless you know your oven really well, to put an oven thermometer in the oven on the rack and adjust accordingly. They're <$10. Try placing it in different spots and figure out how evenly or unevenly your oven heats, and how a pan in one spot affects temperature in another.
Composition and thickness of your pan also matters. Ovens heat from all sides, but it matters whether your food is sitting in glass, steel, thin aluminum, or thick aluminum. Cake mixes try to give different instructions for glass, metal, and dark metal, but it's going to vary by recipe.
And it matters whether you're using a convection or conventional oven. The standard advice is shorter times and lower temperatures for convection, but you might still get differences in terms of drying out the top before the bottom and center cook fully with convection. Maybe you have to cover it part of the time, for some recipes.
If you misjudge and want more crispiness, why not briefly broil at the end? Say you're trying to braise a roast in a pan next to, above, or below the dish of potatoes. Steam from the roast slows the cooking and prevents browning. Then when you take the roast out to rest, you have a couple of minutes to broil before serving.
You would think so, I certainly used to think so, but somehow it doesn't seem to work that way in practice. That's usually the step where my wife does the seasoning and adds the liquids, so IDK if there is something specific she does that makes it work. But I'm definitely whipping them with the whisk attachment, which incorporates air, and not beating them with a paddle attachment. I suspect that's the majority of why it works.