Find the conditional distribution of X=1 given Z=1

Question

Let $X$ follows Bernoulli(1/3), $Y$ independent of $X$ follows Bernoulli(2/3).

$$Z=\begin{cases}X &\text{if $Y=1$}\\ 1-X &\text{if $Y=0$} \end{cases}$$ Find the conditional distribution of $X=1$ given $Z=1$.

I was applying ${P(X=1,Z=1)\over P(Z=1)}$ but not being able to calculate the denominator and numerator help

I am getting $P(Z=1)=4/9 $ And the numerator as $2/9$

Answer 1

Here is the joint distribution of the vector $(X,Z)$

I suppose you are interested in deriving the distribution of $X|Z=1$

that is a bernulli

$$(X|Z=1)\sim B\left(\frac{1}{3}\right)$$

$P(X=1|Z=1)$ is not rv...it's a conditional probability $=1/3$