On Thu, 31 Jan 2013 14:17:38 -0800 (PST), email@example.com wrote:
>On Wednesday, January 30, 2013 3:14:00 PM UTC, David C. Ullrich wrote: >> On Wed, 30 Jan 2013 00:29:09 -0800, William Elliot <firstname.lastname@example.org> >> >> wrote: >> >> >> >> >There is a fair coin with a different integer on each side that you can't >> >> >see and you have no clue how these integers were selected. The coin is >> >> >flipped and you get to see what comes up. You must guess if that was the >> >> >larger of the two numbers or not. Can you do so with probability > 1/2? >> >> >> >> Of course not. Seeing one side gives you no information about >> >> what's on the other side. > >I don't completely agree with this answer. The concept of a number being selected such that "you have no clue how it was selected" doesn't translate readily into mathematics. > If it's mathematics, you need to specify the procedure or specify the proability space (or set of possible probability spaces) etc.
>Since this isn't mathematics, the best I can do is use intuition and commonsense. Surely, if one of the numbers was > 10 ^ 1000, the most intelligent guess is that the number on the reverse side is smaller.
Why in the world would that be? Most positive integers are larger than that. In fact all but finitely many positive integers are larger.
Seems to me that whatever inside your head makes you think that that's the most intelligent guess hasn't quite internalized the bit about how we have no information on how the numbers were chosen.
> >Here's my algorithm: If the number < 0, I will say that the number on the current face is smallest. if the number > 100, I will say that the reverse number is smallest. Otherwise, I will answer randomly. > >This would probably be better than 50/50. > >Alternatively (and probably a better approach), I note that there are three possible answers: A) Yes, such an algorithm exists, B) No such algorithm exists C) The problem is insufficiently specified. > >You went for answer B but I would go for C. > >Paul Epstein