"Almost all objects have property P" vs. "It is easy to test whether an object has property P" I am interested in any relation between "almost all objects(from a universe) possessing a particular property P" versus "testing whether an object has property P being poly. time decidable".

My guess is that they are completely separate (that is, one doesn't imply the other). Am I missing something?

(Note: Almost all in the sense of probability)

PS: I am not sure whether the tag probability is appropriate here, sorry.
They are separate (assuming $P \ne NP$).  Consider the following property $P(x)$: $x$ is a $2n$-bit string, where either the first $n$ bits are not all zeros, or the last $n$ bits are a yes-instance of 3SAT.  It's clear that testing whether $x$ satisfies $P$ is NP-hard, yet almost all strings satisfy it: the density $\to 1$ as $n \to \infty$.