Paul
Posts:
729
Registered:
7/12/10


Re: Beating the Odds?
Posted:
Feb 1, 2013 1:21 PM


On Friday, February 1, 2013 2:47:43 PM UTC, David C. Ullrich wrote: > On Thu, 31 Jan 2013 14:17:38 0800 (PST), pepstein5@gmail.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 <marsh@panix.com> > > >> > > >> 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. > > > > True. > > > > >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. >
1) Most positive integers are larger, but the context is integers not positives. 2) Imagine that you did see 10^1000. There are two choices  guess that the reverse is larger, or guess that the reverse is smaller. If you disagree so strongly that I'm wrong to guess that the reverse is smaller, please could you explain how large the number would have to be for you to guess that the reverse is smaller? Or would that always be a bad guess? 3) Having researched the correct answer, I'm right about 10^1000 even if you restrict the domain to positives. The answer involves selecting your own probability measure and then randomizing a choice according to your own selection. In practice, if you did that on the domain of the positive reals, it would be most unlikely that you would pick a probability measure such that the integral from 10^1000 to infinity is greater than 0.5.
Paul Epstein

