Logic Question

[thylacine]

First, I'll assume that this isn't actually a logic question since it is posted in the probability forum. I'll also assume that claims of this type are always true, and also that there are no ties in the international buffet rankings. Furthermore, I will assume that all buffets in the world post a truthful sign outside saying, for some positive integer j, that they are rated one the top j buffets in the world. This will imply with certainty that they are in fact rated exactly i^th in the world for some positive integer i that is less than or equal to j.

Now for the detailed calculation. For positive integers i and j, suppose that P(i,j) is the probability that a buffet that is actually rated exactly i^th in the world, will post a sign outside saying that they rated one the top j buffets in the world. So of course P(i,j)=0 whenever j is less than i, and

\Sum_{j=i}^{\infty} P(i,j)=1.

Then the probability that a buffet claiming to be rated one the top j buffets in the world, is actually rated exactly i^th in the world is

P(i,j)/ [ \Sum_{k=1}^{j} P(k,j) ].

I may have intentionally made a mistake in what I just wrote. I'm not sure. You guys don't need to be going to buffets anyway.