If neutrinos are majorana fermions, will it be possible for a beta minus decay to emit a neutrino instead of an antineutrino? If neutrinos are majorana fermions, neutrinos and antineutrinos should be equivalent. As a result, reactions like $n\rightarrow p+e^{-}+\nu$, $\nu+p\rightarrow n+e^{+}$ should be kinematically possible. Are these reactions not observed because the branch ratio is too low, or because neutrinos are too difficult to detect?
 A: The reaction pair you give is possible and has been observed many times: the neutrino produced in $\beta$ decay is an antineutrino (as it is produced together with an electron), and can this produce a positron.  The second reaction is called 'inverse beta decay' and was actually used in the Reines and Cowan neutrino discovery experiment.
What I think you meant to ask is whether, if neutrinos are Majorana particles, one could have $n \to p e^- \nu$
followed by $\nu n \to p e^-$, if you somehow gave the neutrino enough energy, perhaps by starting with a beam of high energy neutrons. This is
forbidden by lepton number conservation, but if neutrinos and antineutrinos are the same thing, surely lepton number goes out of the window?
The answer is that even if neutrinos are Majorana particles the reaction  is still forbidden. Almost. The neutrino produced in the first reaction is a right handed (anti) neutrino.  The neutrino in the second reaction must be left handed. Majorana removes the neutrino/antineutrino distinction but not the left/right difference. Lepton number conservation still holds,  but it's thanks to the handedness of the neutrinos involved.
There is an overlap between left and right handed particles (down to the subtle difference between handedness and chirality) but it is VERY small and vanishes in the limit $v \to c$ - and neutrinos, being almost massless, are very close to that limit. This may be exploitable in neutrinoless double beta decay, which is how experiments are trying to resolve the Dirac/Majorana question.
A: Neutrinos are all but unobservable. When interactions involving them are observed, it would be more accurate to say we infer them from everything else. In particular, whether a neutrino or antineutrino is involved follows from lepton number conservation. But if neutrinos are Majorana fermions, we needn't even go to that much trouble; there would simply be no $\nu$/$\overline{\nu}$ distinction.
