Implication of locally perfectly correlated events

In a relativistic and local theory, if we have two space-like separated events $E_{1}$ and $E_{2}$ plus any physical state the theory associates to systems in the past light-cone of $E_{1}$ , called $\Omega_{1}$ , then one has that: $P(E_{1} | E_{2}, \Omega_{1}) = P(E_{1} | \Omega_{1})$ Let's assume that the two events are perfectly correlated or anti-correlated with one another, then this means that the first term can only assume two values: $P(E_{1} | E_{2}, \Omega_{1}) = (1\quad or \quad 0)$ but then, if locality holds, this also means that: $P(E_{1} | \Omega_{1}) = (1\quad or \quad 0)$ To me, this second requirement amounts to say that the theory can determine with certainty whether or not $E_{1}$ is going to happen based on the physical state $\Omega_{1}$ . This seems to suggest that if one has a pair of perfectly correlated or anti-correlated events in a local theory, they each must be separately pre-determined by the physical state in their respective past light-cone. Is this conclusion true or is this reasoning flawed?