Theoretical Random Number Game

[Jazza]

The is between 2 players. Player A picks a number, 0 or 1, Player B tries to guess this number, winner wins say $1 from the loser. They repeat this many times.

Can player A choose his numbers in a non completely random way, such that player B cannot increase his EV above 0?

If you were player B and suspected he was choosing numbers in a not fully random way, what kind of algorithm would you use to try and guess the right number (using his past history of chosen numbers)?

One of the problems I see already is I don't know how to precisly define "a non completly random way"