Consider the following scenario:
Suppose on some date $D1$ the number $N$ is a winning number in a fair lottery where a "play" is considered the selection of a finite set of numbers. By "fair" I mean that the winning number will be selected at random. At some later date $D2$ another lottery, using the same rules, will be held. Suppose one picks the same number $N$ that was the winning number on $D1$ to play on $D2$. Does this selection increase/decrease or have no effect on ones chances of winning the lottery on $D2$?
I believe that picking a previously winning number will have no impact on one's chance of success for the simple reason that the universe has no way of remembering which previous numbers won and which ones did not; since the selection is random, each number has an equally likely chance of being picked regardless of whether it was picked before. Other than a basic assumption of causality, I really don't know though how one would rigorously prove this.
The counterargument, which I believe is faulty, argues against "reusing" winning numbers because the likelihood of the same number coming up twice is infinitesimally small so, of course, one should not reuse numbers. The problem with this as I see it though is that the probability of picking any two specific numbers, regardless of whether they are the same, is identical to picking the same number twice. The fault is that picking numbers this way is like trying to play two lotteries in succession - which is very different from the given problem.