MMSE estimator using pmf table

Question

We have the probability mass function as follows, $$\begin{array}{c|c|c|} & \text{Y=0} & \text{Y=1} \\ \hline \text{X=0} & \frac 15 & \frac 25 \\ \hline \text{X=1} & \frac 25 & 0 \\ \hline \end{array}$$ I want to calculate the $\hat x_M$ MMSE estimator of $X$ given $Y$, and its MSE (Minimum-squared-error). I know $\hat x_M=E[X|Y=y]$. I tried by calculating $E[X|Y=y]$, which is equal to $\sum x{\rho_{X|Y}(x|y)}=\sum x\frac{\rho_{X,Y}(x,y)}{\rho_{Y}(y)}$, and obtained $E[X|Y=0]=\frac{2}{3}$ and $E[X|Y=1]=0$. But I don't know how to continue.