How does the ratio of hydrogen to helium help prove the big bang theory? From what I have read one of the key pieces of evidence for the big bang theory is the ratio of hydrogen to helium in the universe. I have not seen any explanation as to how the big bang theory predicted this ratio (3:1 by mass). 
How did the big bang theory make this prediction?
It seems like the prediction made by the big bang theory would explain why there is more hydrogen than helium, but not how much more. 
If the answer is based on our knowledge of the temperature and size of the universe in the first few fractions of seconds after the big bang, how do we know the time frame of those events in the first place?   
 A: Cosmologists think they understand how the scale factor of the universe has grown with time, based on the Friedmann equations. They also think they understand how the temperature of the universe has cooled as the universe expanded. Finally, they can measure the current temperature of the universe by measuring the frequency distribution of the cosmic microwave background radiation.
So, knowing the current temperature and using equations to extrapolate backwards in time, they come up with temperatures like one billion K at 10 seconds after the Big Bang. This corresponds to a modest energy per particle of 100 keV. I say “modest” because nuclear physicists understand very well how nuclei behave at such energies; nuclear physicists have collided nuclei at such energies for decades.
So the reactions involved in nucleosynthesis have been studied in the laboratory, and the cross section for, say, a proton and a neutron to form deuterium, and the cross section for deuterium and a proton to form Helium-3 are known.
The last piece of the puzzle is knowing the baryon density at that time after the Big Bang. This is extrapolated backward from the current observed baryon density.
If you know the density of protons and neutrons and the cross sections for nuclear reactions, you can do a computer simulation that predicts the abundances of the elements formed during Big Bang nucleosynthesis.
Any uncertainty about the Big Bang is about what happened in the very early universe, say in the first trillionth of a trillionth of a second. But this is way before the time of nucleosynthesis. By the time the universe was baking helium, its physics is believed to be well-understood.
A: In addition to G. Smith's answer, it's worth noting the importance of the neutron-proton ratio. 
Prior to Big Bang Nucleosynthesis (BBN)
Two key nucleon reactions before BBN commences are: 


*

*a neutron interacts with a positron to create a proton plus
antineutrino (and vice versa), and  

*a neutron interacts 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 protons and neutrons. However, as the temperature rapidly dropped, the neutron-proton inter-conversion rate per nucleon fell faster than the Hubble expansion rate, favouring protons ahead of neutrons. About one second after the BB, the temperature had dropped to about 0.7 MeV, too low for these reactions to continue (the "freeze-out"), by which point the N:P ratio had fallen to 1:6.
Nucleosynthesis begins
BBN was now spluttering into action, and the first multi-nucleon element formed is the simplest: deuterium (one proton plus one neutron). However, the temperature was still too high for deuterium to survive, as the energy of some photons was higher than deuterium's binding energy, and any $^2_1$H would quickly photodissociate. This period is called the "deuterium bottleneck"; but once the temperature had dropped to about o.1 MeV, the $^2_1$H could survive, and as a result there was a sudden and significant increase in the proportion of deuterium atoms present. And as helium-4 has the highest binding energy per nucleon among the lighter elements, almost all this deuterium quickly ended up as $^4_2$He. 
However, free neutrons are unstable and decay with a half-life of 611 seconds, so any neutron that hadn't managed to get to "safety" within an atomic nucleus during this brief 20-minute BBN period was most likely to have decayed into a proton. As a result, the final N:P ratio ended up at 1:7.
N:P ratio predicts He/H ratio
This final 1:7 neutron-proton ratio tells us that for every 14 protons at the end of BBN, there were two neutrons. Since on average those two neutrons found themselves with two protons in a $^4_2$He nucleus, 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 (at around 30KeV, or about 20 minutes after the BB), there were roughly 12 hydrogen atoms for each atom of $^4_2$He.  
[There were also trace amounts of other nucleosynthesis residues: about 0.01% of deuterium and $^3_2$He (helium-3), and one part in 10 billion of $^7_3$Li (lithium-7); and also even tinier amounts of the unstable isotopes $^3_1$H (tritium) and $^7_4$Be (beryllium-7), both of which soon decayed.]
So, this is what our knowledge of physics predicts:
By the end of Big Bang nucleosynthesis, on average 12 out of 13 atoms will be hydrogen (92%) and the remaining atom will be 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 now find.
Even stronger evidence
While the question asks "How does the ratio of hydrogen to helium help prove the big bang theory?", there is even stronger support for the standard BBN model when we look at the lesser "relic" product of that brief burst of nucleosynthesis: deuterium. The production of this isotope is highly sensitive to the primordial baryon abundance, and observations of the Universe's large scale structure and of temperature fluctuations in the cosmic microwave background radiation have tightly constrained that number, enabling a well-bounded prediction for the relic abundance of D. This D/H prediction is in close agreement with the current abundance inferred from observations of low metallicity galaxies, providing even stronger support for the standard model.
Reference
For a more technical review of BBN predictions – including pre-BBN processes, N:P ratios and isotope abundances – see (for example) Primordial Nucleosynthesis in the Precision Cosmology Era (Gary Steigman, 2007).
A: The cosmic abundance of helium reinforces ideas about the temporal evolution of the temperature of the universe. The estimated primordial He/H ratio (found by looking at very low metallicity galaxies) matches the predictions of the hot big bang model and fails to be explained by alternatives such as the steady state theory.
If the universe was once hot enough ($>10^{11}$ K) that neutrons and protons existed in a thermal equilibrium, but subsequently expanded and cooled, then the He/H ratio is largely set by four things. (i) the mass difference between the neutron and proton; (ii) the decay timescale of a free neutron; (iii) the temperature range in which stable deuterium can exist without breaking apart or fusing into helium, (iv) the reaction rates for the protons, neutrons and deuterium nuclei, which depends weakly on the ratio of baryons to photons.
The sequence of events is detailed in What is the reason for the shift in balance between neutrons and protons in the early universe? . In summary: the mass difference between the neutron and proton determines that at the epoch that they fall out of thermal equilibrium, their ratio is about 1:6. All the neutrons would subsequently decay, but for the next 200 s or so, the universe is cool enough for deuterium to form and hot enough for deuterium fusion to form helium. During this time a few more free neutrons decay, driving the final n:p ratio to 1:7 by the time the universe cools below the temperature required for further nuclear reactions.
The big bang model perfectly explains these results with only one free parameter - the ratio of baryons to photons, which determines (weakly) the exact fraction of free neutrons that end up forming deuterium nuclei (and then helium) as opposed to freely decaying. However, the primordial deuterium abundance also depends on this parameter in a different (and more sensitive) way. That the estimated primordial He/H and D/H  can be reproduced at with the same baryon to photon ratio is what strongly supports the big bang model.
