On Overcoming Bias, Eli says:
I don’t think you could get up to 99.99% confidence for assertions like “53 is a prime number”. Yes, it seems likely, but by the time you tried to set up protocols that would let you assert 10,000 independent statements of this sort – that is, not just a set of statements about prime numbers, but a new protocol each time – you would fail more than once. Peter de Blanc has an amusing anecdote on this point, which he is welcome to retell in the comments.
Here’s the anecdote:
Conversation with squallmage at 2006-07-28 23:46:54 on OneTrueCalculus (aim)
(23:47:19) OneTrueCalculus: oi
(23:47:23) SquallMage: oi
(23:47:33) OneTrueCalculus: how likely would you rate it that 81,241 is prime?
(23:47:59) SquallMage: very
(23:48:03) OneTrueCalculus: you can go ahead and calculate it if you want, or just give me a probability
(23:49:03) OneTrueCalculus: obviously this would be the subjective sort of probability
(23:49:11) SquallMage: yes.
(23:50:30) OneTrueCalculus: well?
(23:50:35) OneTrueCalculus: or are you busy calculating?
(23:50:38) SquallMage: I said ‘very’.
(23:50:42) OneTrueCalculus: ah
(23:50:46) SquallMage: Quantify that if you must, I’m too tired to.
(23:50:55) OneTrueCalculus: ok
(23:51:10) OneTrueCalculus: I won’t quantify it for you, since I don’t know how you are calibrated
(23:51:22) OneTrueCalculus: okay, how about 7?
(23:51:46) SquallMage: the probability of it’s primacy, or as a quantification of ‘very’?
(23:51:55) OneTrueCalculus: the probability of 7 being prime
(23:52:02) SquallMage: 7 is prime.
(23:52:08) OneTrueCalculus: with probability 1?
(23:52:18) SquallMage: Yes.
(23:52:41) OneTrueCalculus: so will you accept this deal? If you ever find out that 7 is not prime, you will give me $100.
(23:53:10) SquallMage: Only if you explain to me in detail what brought you to propose that deal to me.
(23:53:37) OneTrueCalculus: I am trying to swindle you out of your cash and/or teach you a valuable lesson
(23:54:09) OneTrueCalculus: also, I am trying to figure out how overconfident people are
(23:55:42) SquallMage: Well. I would make a deal with you that I would give you $100 if it were ever proven to me that it was possible in base-ten to generate the quantity 7 by multiplying together any two integers other than 1 and 7.
(23:55:55) OneTrueCalculus: okay
(23:56:05) OneTrueCalculus: I am only talking about the standard natural numbers. No weird groups or anything
(23:56:20) OneTrueCalculus: base 10
(23:56:24) SquallMage: No, ‘7 and 1′ is not different than ‘1 and 7′ also.
(23:56:32) OneTrueCalculus: of course not
(23:56:39) SquallMage: Just checking.
(23:56:42) OneTrueCalculus: I wouldn’t use a cheap technicality like that
(23:56:44) SquallMage: Since I’m a language bastard.
(23:56:49) SquallMage: And I completely would.
(23:56:54) OneTrueCalculus: okay
(23:57:13) OneTrueCalculus: If you even honestly feel that it was a cheap technicality, I wouldn’t expect you to pay me.
(23:57:22) SquallMage: Anyways.
(23:57:23) OneTrueCalculus: Under those conditions, will you accept my offer?
(23:57:33) SquallMage: Yes.
(23:57:37) OneTrueCalculus: Okay.
(23:57:44) SquallMage: Tell me how I owe you $100 now
(23:57:48) OneTrueCalculus: you don’t
(23:57:52) SquallMage: Good.
(23:57:56) OneTrueCalculus: now
(23:58:03) OneTrueCalculus: will you accept the same offer, but for 11 this time?
(23:58:05) SquallMage: Not for 81241
(23:58:34) OneTrueCalculus: i.e. if you ever find out that 11 is not prime, you will give me $100
(23:58:49) SquallMage: I would make that deal under equivalent conditions
(23:59:00) OneTrueCalculus: okay. I’m asking you to make that deal.
(23:59:44) SquallMage: I did say that I would.
(00:00:00) OneTrueCalculus: okay, thanks
(00:00:04) OneTrueCalculus: how about 13?
(00:00:20) SquallMage: Yes.
(00:00:48) SquallMage: 17 also, and 19, and 23.
(00:00:54) OneTrueCalculus: thanks.
(00:00:57) OneTrueCalculus: What about 27?
(00:01:02) SquallMage: No thanks.
(00:01:05) OneTrueCalculus: 29?
(00:01:11) SquallMage: sure.
(00:01:13) OneTrueCalculus: 31?
(00:01:17) SquallMage: Yep.
(00:01:20) OneTrueCalculus: 33?
(00:01:24) SquallMage: Nah.
(00:01:26) OneTrueCalculus: 37?
(00:01:29) SquallMage: Yah.
(00:01:31) OneTrueCalculus: 39?
(00:01:35) SquallMage: Nah.
(00:01:37) OneTrueCalculus: 41?
(00:01:42) SquallMage: Yah.
(00:01:45) OneTrueCalculus: 43?
(00:01:51) SquallMage: Yah.
(00:01:54) OneTrueCalculus: 47?
(00:02:06) SquallMage: Yah.
(00:02:09) OneTrueCalculus: 49?
(00:02:17) SquallMage: Nah.
(00:02:20) OneTrueCalculus: 51?
(00:02:28) SquallMage: Yah.
(00:02:36) OneTrueCalculus: Thank you. I win.
(00:02:48) SquallMage: You know I’ve been up since this time yesterday.
(00:02:51) OneTrueCalculus: You can donate your money to the Singularity Institute for Artificial Intelligence
(00:03:06) SquallMage: Iâ€™ll forward you their message of receipt.