  We study 2-player turn-based perfect-information stochastic games with
countably infinite state space. The players aim at maximizing/minimizing the
probability of a given event (i.e., measurable set of infinite plays), such as
reachability, B\"uchi, omega-regular or more general objectives.
  These games are known to be weakly determined, i.e., they have value.
However, strong determinacy of threshold objectives (given by an event and a
threshold $c \in [0,1]$) was open in many cases: is it always the case that the
maximizer or the minimizer has a winning strategy, i.e., one that enforces,
against all strategies of the other player, that the objective is satisfied
with probability $\ge c$ (resp. $< c$)?
  We show that almost-sure objectives (where $c=1$) are strongly determined.
This vastly generalizes a previous result on finite games with almost-sure tail
objectives. On the other hand we show that $\ge 1/2$ (co-)B\"uchi objectives
are not strongly determined, not even if the game is finitely branching.
  Moreover, for almost-sure reachability and almost-sure B\"uchi objectives in
finitely branching games, we strengthen strong determinacy by showing that one
of the players must have a memoryless deterministic (MD) winning strategy.
