The first is not well defined, the second is and as long as we are talking countable infinity, it is not. (In the uncountable case you are running into problems

The first is not well defined, the second is and as long as we are talking countable infinity, it is not. (In the uncountable case you are running into problems with definitions again.) As a matter of fact, you will not hit almost all numbers. However, you will almost surely have picked a point in any fixed interval, and that is meant with order reapers over long enough timescales. For illustration, you are almost guaranteed to not win the lottery, but when you play again and again, then after something like 14 million times playing (for the German Lotto) you will win, and again after something like 28 million times playing, and so on, so you expect to often over long enough timescales, and you can start to impose additional constrains, for example winning with consecutive numbers, or the first few primes. And after a long enough timescale you start to win again with these additional constrains.