Two street game.

The two players are X and Y. Each start with
equal size stacks S. Each ante A > 0. On the
first street no cards are dealt. On the 2nd
street each are dealt a card from [0,1]. The
player with the lower card wins. A/2 goes to
the house.
1st street. Y may bet any amount S-A. X may
call or fold.
2nd street. No betting. Best hand is awarded
the pot.
Optimal strategy on the 1st street must be
for X to call any bet. X is 50/50 to win
the game. X loses A if he folds and gets
better than one to one to call and play.

This demonstrates that optimal strategy is
not necessarily a winning strategy. It's
just a lose minimum strategy. As we learned
from the movie War Games not playing is
often the best strategy.

In hold'em the streets are not independent.
But they are not linearly dependent either.
There is only a positive correlation among
the streets. Being ahead on one street
increases one chances of being ahead on the
next street. But there are no guarantees.

That is another reason why optimal strategy
on the flop and turn may not always be the
desirable play option. While optimal strategy
exists on the flop and turn, it may not be
the most desirable +EV action.