Measure
$ |\psi\rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \Leftrightarrow \langle \psi | = \begin{bmatrix} \alpha^* & \beta^* \end{bmatrix} $
$ (a + bi)^* = a - bi $
$ |a + bi|^2 = (a + bi)(a + bi)^* = (a + bi)(a - bi) = a^2 + b^2 $
Probability of measuring $ | 0 \rangle $ when in $ | + \rangle $:
$ | \langle 0 | + \rangle |^2 = | \langle 0 | | + \rangle |^2 = | \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix} |^2 = (\frac{1}{\sqrt{2}})^2 = \frac{1}{2} $
Probability of measuring $ | 1 \rangle $ when in $ | + \rangle $:
$ | \langle 1 | + \rangle |^2 = | \langle 1 | | + \rangle |^2 = | \begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix} |^2 = (\frac{1}{\sqrt{2}})^2 = \frac{1}{2} $
Probability of measuring $ | 0 \rangle $ when in $ | - \rangle $:
$ | \langle 0 | - \rangle |^2 = | \langle 0 | | - \rangle |^2 = | \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{bmatrix} |^2 = (\frac{1}{\sqrt{2}})^2 = \frac{1}{2} $
Probability of measuring $ | 0 \rangle $ when in $ | - \rangle $:
$ | \langle 1 | - \rangle |^2 = | \langle 1 | | - \rangle |^2 = | \begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{bmatrix} |^2 = (-\frac{1}{\sqrt{2}})^2 = \frac{1}{2} $