Consider a special case of the experiment: One in which the number picker always picks numbers within a finite interval; i.e., for numbers E1 and E2 inside the envelopes |E1 - E2| < X for some X. In this case, case 3 occurs with zero probability. There is an infinite interval over which both E1 and E2 are greater than any random number R. Similarly, there is an infinite interval over which both E1 and E2 are less than any given random number R. But there is only a finite interval over which it is possible for E1 and E2 to be on opposite sides of R.
Here's another way to think of it: You have an infinitely large dartboard. The author wants to paint a finite sized bullseye on the dartboard. The size of the dartboard is the distance between E1 and E2 (the two numbers in the envelopes). He throws a random dart at the dartboard and if he hits the bullsye he wins. The size of that bullseye is the "advantage" over random chance. But it doesn't matter how large you paint the bullseye, if it's a finite size then there's a zero probability that a random (finite sized) dart will hit the bullseye. The ratio of the area outside the bullseye to the ratio inside the bullseye is infinity.