"...it would often roll a 6, sometimes roll a number adjacent to 6..."
Assuming standard probability applied to my three dice, the odds of my rolling at least one 6 are 1 - (5/6)^3 or approximately 0.4. Assume that the "trick" die rolls a 6 half the time. (Remember I was watching as my opponent also rolled 5's, 4's, and 3's.) Then the probability that I would win a battle is at least 0.4 x 0.5 = 0.2. The attacker odds are actually higher since the attacker would usually win if the defender rolls anything but a 6. My estimate is that with the trick die, the defender would win with frequency around 0.6. So the probability that the defender would win 24 battles is around 0.6^24 or about 1-in-100,000.
"And yes, it's unlikely your friend would (a) weight your dice, (b) waste this ability on a meaningless game of risk, and (c) keep up the act all these years, but it's not 1-in-100-billion unlikely."
There is also (d), even with a "trick" die the event would only be expected to happen with frequency 1-in-100,000. Now combine that low probability with the low probabilities of (a), (b), and (c) also being true. I agree that it is more likely that (a), (b), (c), and (d) are all true is more likely than that a 1-in-100-billion event happened. However, I'm not claiming a 1-in-100-billion event happened. I'm claiming that it is more likely that something unknown occurred, i.e., I have no scientific explanation for the event.
I have no scientific explanation for the event.
Yes, you do: all four dice were weighted. You did your math assuming only one of them was weighted, but if they all were then the event you saw wasn't unlikely at all. Assume that a weighted die rolls the side that it favors with probability p, each of the sides adjacent to it with probability (1-p)/4, and never rolls the side opposite the favored side. How strongly weighted do the dice have to be (that is, what should p be) for 26 consecutive victories for the defender are assured?
The defender automaticall...
What do you believe that most people on this site don't?
I'm especially looking for things that you wouldn't even mention if someone wasn't explicitly asking for them. Stuff you're not even comfortable writing under your own name. Making a one-shot account here is very easy, go ahead and do that if you don't want to tarnish your image.
I think a big problem with a "community" dedicated to being less wrong is that it will make people more concerned about APPEARING less wrong. The biggest part of my intellectual journey so far has been the acquisition of new and startling knowledge, and that knowledge doesn't seem likely to turn up here in the conditions that currently exist.
So please, tell me the crazy things you're otherwise afraid to say. I want to know them, because they might be true.