Let's assume we have two discrete probability spaces $(\Omega_1,P_1)$ which describes experiment $1$ and $(\Omega_2,P_2)$ which describes experiment $2$. We know the probability $P_1$ but we don't know exactly how $P_2$ looks like. For a set $A\subset \Omega_2$ we know that $P_2(A)=\frac{1}{3}$. Now we find out that $A\subset \Omega_1$ and that for all $\omega\in A$ experiment $2$ is the same as experiment $1$.
E.g. experiment $1$ is a negative binomial experiment and all $\omega\in A$ also satisfy the conditions of the negative binomial experiment.
Does this imply $P_1(A)=P_2(A)=\frac{1}{3}$ or $P_1(\omega)=P_2(\omega)$, $\forall \omega\in A$? If yes, why? If not, why?