Here's some stuff from the lecture dealing with the theory of the big bang:

Based on these formulas, we can calculate how many atoms there are in the universe.

This is the formula for density:

the density of normal matter in the universe as 5 x 10^-28/a(t)^3 kg m^-3

a typical atom has mass of 2 x 10^-27 kg

And based on that, to calculate roughly how many atoms there are in the universe today per cubic meter, we get 1 atom per 4 cubic meter.

Now my question is how accurate is this considering we don't know how large the universe is or its approximate dimensions? How do we know the estimate coordinates of a galaxy from billions of years ago using the scale factor? I do understand how distances are being calculated by the redshift that we see through the telescope, but I don't see how we can be sure how many atoms there are.

Though assuming it is correct, how do we determine the actual size of the universe that we can see today?