In post 926, Varsoon wrote:@S_S: That'd be the case if Arthur knew he was Arthur, but he doesn't, so if Arthur doesn't attempt to get the sword, his chances are always 0. Since no one knows if they are arthur or not other than Merlin, everyone should try to get sword, right?
Let's say that N townies (out of 12) attempt to get the sword. The odds that Arthur is one of them are N/12. (There are 12 possibilities for Arthur and N of them are going for the sword.)
Then, the odds that Arthur, assuming he does attempt it, is the one who actually gets it are 1/N. (There are N people going for the sword and 1 of them can get it.)
The odds that Arthur ends up with the sword, assuming at least one person attempts it, are (N/12) * (1/N) = 1/12 regardless of what N is.
Another way to look at it: exactly one townie is gonna end up with the sword. No matter how they ended up with it, there are 12 identical townies and one of them is Arthur, so if you pick a random townie to be Arthur the odds that you picked the one with the sword are 1/12.