Magic: the Gathering comes to mind, since the community often focuses explicitly on thinking-about-probabilities due to both the hidden information (what cards does your opponent have?) and the random factor (what cards will I draw later in the game?)

One of my favorite stories is about a high-level competitor who had a tendency to win since he could work out "I will ONLY win if I draw this one card" and would then craft his entire game around the expectation of that card - after all, if he failed to draw it, he was dead anyway :)

When I wrote the comment, I explicitly thought of poker, chess, the Settlers of Catan and Wizard, all of these games where thinking with probabilities (w.r.t. the opponents' cards / the opponents' actions and the state of the board / the opponents' actions and the roll of the dice / the opponents' cards, respectively) can be more or less helpful. In the end, I chose diplomacy because additionally, it shares the characteristic of simultaneous moves.

There are a large number of games where the probabilistic nature is more readily apparent. Bridge is one of my favorites, but poker and backgammon both come to mind.