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A physical system has dynamical variables $X,Y,Z$ represented by the
$2\times2$ Pauli matrices $\sigma_x,\sigma_y,\sigma_z$.
Compute the average values of the following operators in the specified
states.
- The average of $X$ in the state represented by the vector $\begin{pmatrix}1 \\ 2\end{pmatrix}.$
- The average value of $Y+Z$ in the state in which the variable $X$ has a definite value -1 for $X$.
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