[QUE/QM-06013]

The vector space needed to describe a particular physical system is two dimensional complex vector space. The  states are therefore represented by 2 component complex column vectors. The observables for this system are $2\times2$
matrices. A set of three dynamical variables of the system, $X,Y,Z,$ are be represented by $2\times 2$ matrices denoted by $\sigma_x, \sigma_y,\sigma_z$, where $$  \sigma_x =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad \sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$  If the state vector of the system is given by $$   \left( \begin{array}{r} \displaystyle{1\over\sqrt{5}}\\[4mm] \displaystyle {-2\over\sqrt{5}} \end{array} \right) $$ Find the probability that

1. a measurement of $X$ will give value $1$.
2. a measurement of $Y$ will give value $-1$.
3. a measurement of $Z$ will give value $1$.
4. a measurement of $X+Y$ will give value $\sqrt{2}$.