Multiple points, really. I believe that this calculation is flawed in specific ways, but I also think that most calculations that attempt to estimate the relative odds of two events that were both very unlikely a priori will end up being off by a large amount. These two points are not entirely unrelated.
The specific problems that I noticed were:
And then the meta-problem: when you're multiplying together more than two or three probabilities that you estimated, particularly small ones, errors in your ability to estimate them start to add up. Which is why I don't think it's usually worthwhile to try and estimate probabilities like this.
But you have a fair point about it being a good idea to practice explicit calculations, even if they're too complicated to reliably get right in real life. So here's how I might calculate it:
P(bear encounters you): 1%.
P(tent scratched | bear): 60%, for the reasons I said above... unless we take into account it scaring away other tent-scratching animals, in which case maybe 40%.
P(tent flipped over | bear & tent scratched): 20%, maybe? I think if the bear has already taken an interest in your tent, it's more likely than usual to flip it over.
P(you see a bear-shaped object | bear & tent scratched & tent flipped over): Bears always look like bears. This is so close to 100% I wouldn't even normally include it in the calculation, but let's call it 99.99%.
P(you get eaten | bear & tent scratched & tent flipped over & you see a bear-shaped object): It's already pretty been aggressive so far, so I'd say perhaps 5%.
On the other side, there are almost no objects for which the probability of it looking exactly like a bear isn't infinitesimal; let's only consider Bigfoot and serial-killer-who's-a-furry for simplicity, then add them up.
P(Bigfoot exists): ...hmm. I am not an expert on the matter, but let's say 1%.
P(Bigfoot encounters you | Bigfoot exists): There can't be that many Bigfoots (Bigfeet?) out there, or else people would have caught one. 0.01%.
P(tent scratched | Bigfoot): Bigfeet are probably more aggressive than bears, so 70%.
P(tent flipped over | Bigfoot): Again, Bigfeet are supposed to be pretty aggressive, so 50%.
P(you see a bear-shaped object | Bigfoot & tent scratched & tent flipped over): Bigfoot looks similar enough to a bear that you'll almost certainly think he's a bear. 99%.
P(you get eaten | Bigfoot & tent scratched & tent flipped over & you see a bear-shaped object): Again, Bigfeet aggressive, 30%.
Then for the furry cannibal one:
P(furry cannibal stalking this forest): 0.000001% (that's one in a hundred million, if I got my zeroes right). I welcome you to prove me wrong on the matter by manually increasing the number of furry cannibals in a given forest.
P(furry cannibal encounters you | furry cannibal exists): How large of a forest is this? Well, he probably has his methods of locating prey, so let's say 10%. Wait, why did I assume he's a "he"? What gender is the typical furry cannibal? Probably a trans woman? Let's name this furry cannibal Susan.
P(tent scratched | Susan): Probably not that high; she doesn't want to wake you up too soon. 30%.
P(tent flipped over | Susan & tent scratched): She might just sneak in, but let's say 90%.
P(you see a bear-shaped object | Susan & tent scratched & tent flipped over): She's wearing a bear costume, as hypothesized; 99.99%.
P(you get eaten | Susan & tent scratched & tent flipped over & you see a bear-shaped object): Yes, of course this happens; this was her whole kink in the first place! 99%.
So for "bear," we have 1%*40%*20%*99.99%*5% = 0.004%. For "Bigfoot," we have 1%*0.01%*70%*50%*99%*30% = 0.00001%. For "Susan," we have 0.000001%*10%*30%*90%*99.99%*99% = .000000027%. Looks like Bigfoot was so much more likely than Susan that we can pretty much just forget the Susan possibility altogether. It's 0.004 to 0.00001, so 400 to 1 chance that you're being eaten by a bear.
(Although I actually think you should be even more confident than 400 to 1 that it's a bear rather than Bigfoot, and that I just was off by an order of magnitude for one reason or another, as happens when you're doing these sorts of calculations. And if you ever actually observe all of these things, the most likely hypothesis is that you're dreaming.)
You can just try to estimate the base rate of a bear attacking your tent and eating you, then estimate the base rate of a thing that looks identical to a bear attacking your tent and eating you, and compare them. Maybe one in a thousand tents get attacked by a bear, and 1% of those tent attacks end with the bear eating the person inside. The second probability is a lot harder to estimate, since it mostly involves off-model surprises like "Bigfoot is real" and "there is a serial killer in these woods wearing a bear suit," but I'd have trouble seeing how it could be above one in a billion. (Unless we're including possibilities like "this whole thing is just a dream" - which actually should be your main hypothesis.)
In general, when you're dealing with very low or very high probabilities, I'd recommend you just try to use your intuition instead of trying to calculate everything out explicitly.* The main reason is this: if you estimate a probability as being 30% instead of 50%, it won't usually affect the result of the calculation that much. On the other hand, if you estimate a probability as being 1/10^5 instead of 1/10^6, it can have an enormous impact on the end result. However, humans are a lot better at intuitively telling apart 30% from 50% than they are at telling apart 1/10^5 from 1/10^6.
If you try to do explicit calculations about probabilities that are pretty close to 1:1, you'll probably get a pretty accurate result; if you try to do explicit calculations about probabilities that are several orders of magnitude away from each other, you'll probably be off by at least one order of magnitude. In this case, you calculated that even if a person on a camping trip is being eaten by something that looks identical to a bear, there's still about a 2.6% chance that it's not a bear. When you get a result that ridiculous, it doesn't mean there's a nonbear eating you, it means you're doing the math wrong.
*The situations in which you can get useful information from an explicit calculation on low probabilities are situations where you're fine with being off by substantial multiplicative factors. Like, if you're making a business decision where you're only willing to accept a <5% chance of something happening, and you calculate that there's only a one in a trillion chance, then it doesn't actually matter whether you were off by a factor of a million to one. (Of course, you still do need to check that there's no way you could be off by an even larger factor than that.)
It doesn't matter how often the possum would have scratched it. If your tent would be scratched 50% of the time in the absence of a bear, and a bear would scratch it 20% of the time, then the chance it gets scratched if there is a bear is 1-(1-50%)(1-20%), or 60%. Unless you're postulating that bears always scare off anything else that might scratch the tent.
Also, what about how some of these probabilities are entangled with each other? Your tent being flipped over will almost always involve your tent being scratched, so once we condition on the tent being flipped over, that screens off the evidence from the tent being scratched.
Also, only 95% chance a bear would look like a bear? And only 0.01% chance it would eat you?
Realistically, once we've seen a bear-shaped object scratch your tent, flip it over, and start eating you, you should be way more confident than 38 to 1 that you're being eaten.
"20% a bear would scratch my tent : 50% a notbear would"
I think the chance that your tent gets scratched should be strictly higher if there's a bear around?
Do you have any specific examples of what this new/rebooted organization would be doing?
It sounds odd to hear the "even if the stars should die in heaven" song with a different melody than I had imagined when reading it myself.
I would have liked to hear the Tracey Davis "from darkness to darkness" song, but I think that was canonically just a chant without a melody. (Although I imagined a melody for that as well.)
...why did someone promote this to a Frontpage post.
If I'm understanding correctly, the argument here is:
A)
B)
C)
Therefore, .
First off, this seems to have an implicit assumption that .
I think this assumption is true for any functions f and g, but I've learned not to always trust my intuitions when it comes to limits and infinity; can anyone else confirm this is true?
Second, A seems to depend on the relative sizes of the infinities, so to speak. If j and k are large but finite numbers, then if and only if j is substantially greater than k; if k is close to or larger than j, it becomes much less than or greater than -1/12.
I'm not sure exactly how this works when it comes to infinities - does the infinity on the sum have to be larger than the infinity on the limit for this to hold? I'm pretty sure what I just said was nonsense; is there a non-nonsensical version?
In conclusion, I don't know how infinities work and hope someone else does.
I think I could be a good fit as a writer, but I don't have much in the way of writing experience I can show you. Do you have any examples of what someone at this position would be focusing on? I'm happy to write up a couple pieces to demonstrate my abilities.
Had a dream last night in which I was having a conversation on LessWrong - unfortunately, I can't remember most of the details of my dreams unless I deliberately concentrate on what happened as soon as I wake up, so I don't know what the conversation was about.
But I do remember that I realized halfway through the conversation that I had been clicking on the wrong buttons - clicking "upvote" & "downvote" instead of "agree" and "disagree", and vice versa. In my dream, the first and second pairs of buttons looked identical - both of them were just the < and > signs.
I suggested to the LW team that they put something to clarify which buttons were which - maybe write the words "upvote", "downvote", "agree", and "disagree" above the buttons. They thought that putting the words there would look really ugly and clutter up the UI too much.
But when I woke up, it turned out that the actual site has a checkmark and an X for the second pair of buttons! And it also displays what each one means when you hover over it! So thanks for retroactively solving my problem, LW team!