What formula is used to calculate the age of the universe?

Use Hubble constant relationship to time, i.e.:

#t = 1/H_o#

substitute an estimate for Hubble constant #75 (km)/s/(Mpc)#, and knowing that #=>1Mpc= 3.09xx10^19 km#

#t = 1/(75(km)/s)*(3.09xx10^19 km)=4.12xx10^17 s#

convert to years

#t = 4.12xx10^17 s * 1/60min/s*1/60(hr)/min*1/24 (day)/(hr)*1/365(yrs)/(day)#

#t= 1.308xx10^10#

but be carefully there are 5 cases:

1) Non-zero cosmological Constant (yes that Einstein cosmos constant): #t > 1/H_o#
2) Empty Universe model (no mass): #t = 1/H_o#
3) Open: #t ge 1/H_o > 2/3H_o#
4) If the universe has exactly the right amount of mass, it will neither stop expanding, nor re-contract either. This model is referred to as the “critical universe”: #t =2/3 1/H_o#
5) Open Model: #t < 2/3 1/H_o#

Typically what you will hear is the universe is 13.8 billion years old, that is #13.8xx10^9# clearly this tends to Empty or Non-zero cosmological Constant Universe.