There are two players $A$ and $B$ that play the following game. Each of them is given a random positive integer number by a fair judge, and the player with the biggest number wins. The judge tells player $A$ his number then waits some time(let's say 1 minute) and then tells player $B$ his number.
Obviously the probabilities of winning are $50-50$ (equal). But after the judge tells player $A$ his number (let's name it $x$), one could assume (no matter what value the $x$ has) that the probability of $B$ winning is $1$, since there are infinite integers greater than $x$ but only $x-1$ positive integers smaller than $x$(and the probability of draw is $0$).
I know that the game must be fair, but why does the information we take from hearing player's $A$ number condemn player $A$ to lose? Why do we(me at least) think for 1 minute that player $B$ will surely win?
It seems like a paradox to me and I would like an explanation/solution for it. I thought of this while reading about another paradox (Necktie paradox) but I can't give myself an explanation for it. What is the flaw in the "logic" I used?