I also did the math, but got a slightly different result:
My approach: "how many uncommon do you put in, how many uncommon and rare do you get out on 10 runs of 3/4/5 uncommon inputs with perfectly average luck"
5: 10*5 uncommon in, 10 win/0 loss so 10 rare out.
10 rare / 50 uncommon = 0.2 rare/uncommon
4: 10*4 uncommon in, 8 win/2 loss so 8 rare out, 2 uncommon out.
8 rare / (40 uncommon - 2 uncommon) = ~0.21053 rare/uncommon
3: 10*3 uncommon in, 6 win/4 loss so 6 rare out, 4 uncommon out.
6 rare / (30 uncommon - 4 uncommon) = ~0.23077 rare/uncommon
So I get (6/26)/(1/5) = 15/13 = ~ +15.4% rares/uncommon from running sets of 3 instead of 5, not the +13% you got.
It can't be +15.4% and +13%, so there's gotta be a mistake somewhere.
Anyone care to point out the thinko?