I have a Problem with this exercise: $X,Y:\Omega \to \{0,1\}$ are random variables with $X$~Bernoulli($\frac{1}{2}$) and $Y$~Bernoulli($\frac{3}{4}$). We also know that $P(X=Y=0)=\frac{1}{4}$
I already showed that $X$ and $Y$ are not independet. Now I want to determine the distribtion of $XY$ Therefore I have to calculate $\rho(1)=P(X=1 \cup Y=1)-P(X=1 \cap Y=1)=P(X=1)+P(Y=1)-P(X=Y=1)=\frac{1}{2}+\frac{3}{4}-\frac{3}{4}=\frac{1}{2}$ Because $P(X=Y=1)=1-P(X=Y=0)=\frac{3}{4}$ is this true?
Therefore I thought $\rho(0)$ musst be $\frac{1}{2}$ because $\rho(0)+\rho(1)=1$ But If I try to calculate $\rho(1)=P(X=1 \cup Y=0)+P(X=0 \cup Y=1)+P(X=0 \cup Y=0)=P(X=1)+P(Y=0)+P(X=0)+P(Y=1)+P(X=0)+P(Y=0)-P(X=Y=0)$ But thats definitly not right. I hope you can help me
$$P(XY=0)=P((X=0)\cup (Y=0))$$ $$=P(X=0)+P(Y=0) -P(X=Y=0)=\frac 1 2 +\frac 1 4 -\frac 1 4=\frac 1 2.$$
And $P(XY=1)=1-P(XY=0)=1-\frac 1 2 =\frac 1 2$.