Calculate the probability of measuring the electron in a \"spin-down\" state, given the following wavefunctions:
(i) $\vert Y_{30}(\theta, \phi)\alpha\rangle$
(ii) $\vert Y_{32}(\theta, \phi)\beta\rangle$
(iii) $2^{-1/2}(\vert Y_{30}\rangle(\vert \alpha\rangle + \vert \beta\rangle))$
(iv) $5^{-1/2}(\vert Y_{32}\rangle(\vert \alpha\rangle + 2\vert \beta\rangle))$
