Well...
LittleFinger early on in the series spoke of having family from braavos, the home of the faceless men. His Grandfather, I believe. Plus being the master of coin, would frequent the iron bank in braavos so is well aquinted with the area.
LF "likes to play a game...to see the worst outcome blah blah". Bran the 3 eyed Raven, reiterates back to LF his quote, "Chaos is a ladder", spioken to Ned when he betrayed him, their father. I'd assume that this would've given LF a heads up that something is very very wrong, and he'd begiin to make his preperations, LF style.
In one scene, he is talking secretly to a young woman in a barn (or something), she whispers, "your time's up" and he gives her a silver looking coin. you need to have the sound up to hear it. Who was she and why were they talking, especially because she didn't look like anyone of importance but simply a kitchen maid or similar. Nothing else is referenced in later episodes of their meeting, so it's kind of left open..
The silver coin could be an iron one, like the one Ayra gave to the captain to board his ship to braavos, and it could very well have been payment to a faceless assasin to take his place.
LittleFinger is too sneaky to have been caught out as he (apparently) did.