In the context of the wave functions of electron in a hydrogen atom, I just want to get clear regarding the terms probability and probability density of an finding an electron. To be precise, we know that the value of $|\Psi|^2$ at a point gives the probability density of the electron at the point. Now, does that imply that if the value of $|\Psi|^2$ is maximum at a point, the electron is most likely to be found at that point?
Reason for my doubt: The question given by my instructor goes:
Given that the normalized wave fuction of the electron in $\mathrm{3d}_{z^2}$ orbital of an Hydrogen atom is: $$ \Psi = N \sigma^2 \mathrm{e}^{-\frac{\sigma}{3}}(3\cos^2\theta-1) \, , $$ where $\sigma={\frac{r}{a_o}}$
Now where is the electron most likely to be found?
The solution given goes like: If $P$ represents probability of finding the electron and $\mathrm{d}V$ the volume element then, $$ \frac{\mathrm{d}P}{\mathrm{d}V} = |\Psi|^2 \implies \mathrm{d}P=|\Psi|^2 \mathrm{d}V$$ Also since, $$\mathrm{d}V = r^2 \sin\theta \mathrm{d}r \mathrm{d}\theta \mathrm{d}\phi$$ We get, \begin{align} \mathrm{d}P &= |\Psi|^2 r^2 \sin\theta \mathrm{d}r \mathrm{d}\theta \mathrm{d}\phi \\ &= N_1 \sigma^4 \mathrm{e}^{-\frac{2\sigma}{3}}{(3\cos^2\theta-1)^2} \sigma^2 \sin\theta \mathrm{d}\sigma \mathrm{d}\theta \mathrm{d}\phi \\ & = N_1 [ \sigma^6 \mathrm{e}^{-\frac{2\sigma}{3}} \mathrm{d}\sigma ][ {(3\cos^2\theta-1)^2} \sin\theta \mathrm{d}\theta ][ \mathrm{d}\phi] \end{align}
He later maximized the $[\sigma^6 \mathrm{e}^{-\frac{2\sigma}{3}}]$ and $[{(3\cos^2\theta-1)^2} \sin{\theta}]$ terms to get the values of ${\sigma}$ i.e. hence $r$ and ${\theta}$ at which the electron is most likely to be found. Infact, he stressed on the maximization of ${\theta}$ part which came out to be $0$. And hence, a conclusion made that for an electron in $\mathrm{3d}_{z^2}$ orbital, the electron is most likely to be found in the XY plane.
I find all this stuff confused and fishy. Can someone explain me this solution or prove it wrong via a right way of solving this question.