How was it determined how many neutrinos result from a single Beta decay? What experiment, if any, shows how many neutrinos are produced for the decay of one W boson?
 A: I cannot improve on the text I wrote in The Large and the Small, which follows with minor edits.

Beta radiation had been understood as resulting from beta decay of the neutron
  into proton and electron: $n\rightarrow p + e$, but it was found that this violates conservation of energy-momentum (and conservation of spin). In 1931 Wolfgang Pauli proposed that another particle is produced in the decay. Pauli initially referred to the new particle as a neutron, “neutral one”. The name neutrino, meaning “little neutral one” in Italian, was jokingly coined by Edoardo Amaldi in a conversation with Enrico Fermi and was used by Fermi at a conference in Paris in July 1932 to distinguish it from Rutherford’s neutron.
Fermi unified Pauli’s neutrino with Dirac’s treatment of the electron and
  Rutherford’s neutron-proton model of the nucleus, and gave a solid theoretical
  basis for future experimental work. He described the antineutrino as a product
  of the decay of a neutron into proton, electron and antineutrino: $n\rightarrow p + e +\bar \nu$. His paper was rejected by the journal Nature as “too remote from reality” (this is typical of poor standards of peer review at all journals). It was published in an
  Italian journal in 1934, but the general lack of interest in theory at that time
  caused Fermi to switch to experimental physics.
Neutrinos are produced in weak interactions, so called because of the infrequency
  with which they occur. The neutrino has been thought to have zero mass,
  but now we believe it has a very low mass. It interacts with other matter so rarely
  that although large numbers of neutrinos are produced in the Sun, almost all of
  them pass straight through the Earth.
Pauli thought the neutrino would be so hard to detect that he offered a case
  of champagne for its discovery.Frederick Reines and
  Clyde Cowan detected antineutrinos, produced in enormous quantities by the
  Savannah River nuclear reactor in the USA. Reines shared the 1995 Nobel Prize
  with Martin Perl. Reines’ & Cowan’s method was to look for the reverse interaction,
  in which an antineutrino combines with a proton to produce a neutron and
  a positron, . The positron almost immediately combines with an
  electron in the environment, to give a distinctive flash of two gamma photons,
  in opposite directions. The neutron was also detected by absorption by cadmium,
  with the emission of a third gamma photon within 5 microseconds. The
  three gamma photons were detected using a scintillating chemical which produces
  flashes of visible light in response to gamma photons. Reines and Cowan
  were able to detect about three neutrinos per hour, and to confirm the predicted
  difference between the observed rate with the reactor operating and the rate of
  neutrinos from the Sun when the reactor was switched off.
The $Z$ and $W$ were inferred later, as a result of theoretical attempts to describe interactions consistently in a manner similar to the electromagnetic interaction in quantum electrodynamics. The first experimental evidence of the $Z$ boson was the observation of a weak neutral current, in which an electron is kicked into motion by a neutrino without being converted to a different particle, in the Gargamelle bubble chamber at
  CERN in 1973. Neutrinos are almost as likely to scatter by $Z$ boson exchange as
  by $W$ boson exchange. A number of authors, including Glashow, had predicted
  this interaction, but Weinberg also predicted its strength. The $W^\pm$ and the $Z^0$ were found in high-energy experiments in 1983, with masses of 80.4 GeV and
  91.2 GeV respectively, also at CERN, in proton-antiproton collisions using the
  Super Proton Synchrotron.

A: 
In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar. For example, beta decay of a neutron transforms it into a proton by the emission of an electron accompanied by an antineutrino; or, conversely a proton is converted into a neutron by the emission of a positron with a neutrino in so-called positron emission. 

Here is the Feynman diagram for the free neutron decay:

As the kinematics showed a continuous momentum spectrum for the detected positron, it implied missing energy, and fitted a three body decay with a low mass neutral particle. 

Early studies of beta decay revealed a continuous energy spectrum up to a maximum, unlike the predictable energy of alpha particles. Another anomaly was the fact that the nuclear recoil was not in the the direction opposite the momentum of the electron. The emission of another particle was a probable explanation of this behavior, but searches found no evidence of either mass or charge. The interesting history has Wolfgang Pauli in 1930 proposing an as yet unobserved particle to explain the continuous distribution of energy of the emitted electrons. Then Enrico Fermi called this particle a neutrino and developed a theory of beta decay in which the neutrino carried away the missing energy and momentum. With no charge and almost no mass, it was hard to detect, and not until 1956 was experimental detection of the neutrino achieved. For symmetry reasons, the particle emitted along with the electron from nuclei is called an antineutrino. The emission of a positron is accompanied by a neutrino.

The virtual W decays to an electron and an electron antineutrino , needed to preserve lepton number conservation.
The concept of lepton number conservation developed slowly as data was accumulated for the decays of particles. The increase in the energy of accelerator led to the discovery of more elementary particles, that led to the establishment  of the present day standard model.
Lepton number conservation as a law  started being used  with the muon decay seen  in cosmic ray muons. There are decays of the muon that are not observed, which led to the axiomatic definition of lepton number conservation, where the leptons now  are the $τ$ the  $μ$ and the $e$. 
