Oh dear. No, p values are not like an inverse of the probability of a difference being due to chance.
It's much worse than that.
The question we generally want answered is something like 'are these two (or more) things genuinely different or are they really the same?'
The question a p-value answers is 'Assuming these two things are underlyingly identical, and a certain set of parameters describes the distribution of their apparent values, and we repeated this experiment/procedure an arbitrarily large number of times, how many of those repetitions would give us differences at least as extreme as the ones we observe here?'
This reasoning is exactly as tenuous as it sounds.