Why is the neutrino oscillation probability formula independent of the Majorana phases? Is there a simple way to understand why the neutrino oscillation probability formula is independent of the Majorana phases?
 A: I assume you mean the routine formula, for Greek flavor indices and Latin mass eigenstate indices,
$$
P_{\alpha\rightarrow\beta}=\left|\left\langle \nu_\beta(t)|\nu_\alpha\right\rangle \right|^2=\left|\sum_i U_{\alpha i}^{*}U_{\beta i}e^{ -i m_i^2 L/2E }\right|^2.
$$
Now the standard result of Bilenky et al 1980 extends the PMNS matrix to 
$$
U\mapsto U P,
$$
where $P\equiv \operatorname {diag}(1,\exp (i\alpha_{21}/2), \exp(i\alpha_{31}/2))$. One calls these two αs Majorana phases, all the remaining phases of the diagonalized Majorana masses having been absorbed into the conventional-looking "CP-phase" in U. It's just a name.
Now, for this formula, visibly, the sum over i involves, for each i respective factors of  $\exp(-i\alpha_i + -im_i^2 L/2E+i\alpha_i)$, involving
P* , P, and the propagator term. But the respective P* and P terms in the exponents of this diagonal cancel each other, and, presto!, the propagator is not modified, $=\exp(-im_i^2 L/2E)$, and all trace of the two Majorana phases is gone in this particular formula.
Note this is predicated on the highly restricted and peculiar nature of "conventional" neutrino oscillation experiments, that cannot measure real Majorana processes, violating lepton number, $\Delta L=2$, like neutrinoless double β decay --experiments which can do that, like CUORE, can also, in principle, access these extra phases, but are still languishing empty-handed. Bilenky et al, in 1980, also fantasize about possible right-handed extra interactions, etc... that might allow access to these phases in oscillation experiments.
