In addition to G. Smith's answer, it's worth noting the importance of the neutron-proton ratio.
The two key reactions are that a neutron can react with a positron to create a proton plus neutrino (and vice versa), and a neutron can react with a neutrino to create a proton plus electron (and vice versa). Initially the temperature was high enough for all these reactions to take place, maintaining an equilibrium of roughly equal numbers of protons and neutrons. However, as the temperature rapidly dropped, this favoured the creation of protons ahead of neutrons, until by about one second after the BB the temperature had dropped to about 0.7 MeV - the "freeze-out temperature" - at which point the N:P ratio had fallen to 1:6.
The first nucleon to form is the simplest: deuterium (one proton plus one neutron). However, while deuterium could now form, the temperature was still too high for it to survive, as the energy of some photons was higher than deuterium's binding energy. Once the temperature had dropped to about o.1 MeV - the "deuterium bottleneck" - there was a sudden burst of deuterium formation. And as helium-4 has the highest binding energy per nucleon among the lighter elements, almost all the deuterium ended up as He4.
Not all neutrons managed to get to "safety" within a nucleon: free neutrons are unstable and decay with a half-life of 611 seconds, and some of these decayed into protons before they could fuse with a proton. As a result, the final N:P ratio ended up at 1:7.
This ratio tells us that for every 14 protons there were two neutrons. Since on average those two neutrons end up in a He4 nucleus with two protons, this leaves 12 protons without a partner. A proton on its own is a hydrogen ion, so we can predict that once the temperature had dropped too low for further nucleosynthesis (about 10-20 minutes), there were roughly 12 hydrogen atoms for each atom of helium-4.
[There were also trace amounts of other nucleosynthesis residues: about 0.01% of deuterium and helium-3, and one part in 10 billion of lithium-7; and also some tiny amounts of the unstable isotopes hydrogen-3 (tritium) and beryllium-7, which later decayed.]
So, this is what our knowledge of physics predicts: 12 out of 13 atoms are hydrogen (92%) and the remaining atom is helium (8%); or by mass, with a helium atom four times heavier than a hydrogen atom, it's 4:12 or 4/16ths or 25% helium, and 75% hydrogen. And this is, indeed, exactly what we find.