Amazing Fact in Probability! Record Setting and Symmetry of Independence

Consider this scenario:
You've been playing your favorite game for a while (I've been recently enjoying Temple Run!), so your skills have practically leveled off and you're basically just playing the game for fun and to see what's the highest score you can achieve!

Lo and behold, after many hours of playing late into the night you achieve a high score on a particular run! Flushed with success but tired as well, you wonder whether it is worth it try another run or not. Intuitively, one would think that achieving a high score on the next run is going to be even more difficult so the probability of the next one run being a record setting run is lower now that you've just set a record on the current run.

Is this intuition correct? Is the probability of the next run being record setting affected by the fact that the current run is record setting?
Stop and think for a bit...
...
Maybe a bit more...
...
Okay now let's take a closer look.
Another way to look at the question is to turn it around: Is the probability of the current run being record setting affected if it is fated that the next run will be record setting? Now, our intuition would say no, the current run does not care what the next run does; it is independent of the next run.  Here is the amazing fact: since the setting a record on the current run is independent of setting a record on the next run, so too setting a record on the next run is independent setting a record on current run! Independence is symmetric! It doesn't matter whether you just set a record or not!

But be careful, I'm not saying that the chance of setting a record on the next run is the same as setting a record on the current run.  Think of it this way, to be a record, a run must beat out all previous runs. For the current run to be a record, it must beat out all previous runs, but for the next run to be a record, it must beat out all previous runs and the current run. Therefore, setting a record on the next run will indeed have lower probability than setting a record on the current run, but only because it has to beat out more runs, not because it's affected by the outcome of the current run.

My worldview has just been recalibrated.