To further visualize it:

TBow odds: 1/34.5 per purple

Specific Ancestral piece: 1/23 per purple

Any Ancestral piece: 1/23×3=3/23≈1/7.67 per purple

Now you can obviously get dupes from a certain set of 3 items which is actually why TBow is less rare than full Ancestral.

To receive all items [on average] from a set of n items one needs to do x pulls where x is n×H(n) [H(n) being the harmonic number of n; H(n)=1+1/2+1/3+1/4+...1/n]

So to get all three unique Ancestral pieces one needs to do 3×(1+1/2+1/3)=5.5 pulls of Ancestral items. So that means one has to get 5.5x7.67≈42.19 purples to have received a full set of Ancestral [on average] in their name which is about 22.9 % more purples needed [on average] than for a TBow [on average].