Unless you worded it incorrectly then you're wrong
Let's take 3 to cube
If square is times by itself the cubed = 3x3x3
3x3 =9 then x3=27
power of three is cubed
If we also take into account the less common ability then we have
C=Common
N=Normal
R=Rare
CCCCC N N N N N N N N N N N (R)/2
For simplicity i'll double each however the proportions are the same so there is no difference in the final calculation.
CCCCCCCCCC NN NN NN NN NN NN NN NN NN NN NN R
So we have a 10/33 of C approx:30%
2/33 per normal ability approx:6%
1/33 for the rare ability approx:3%
Now repeat that process for the next slot:
30%*30%=0.3*0.3=0.09=9%
6%*6%=0.36%
3%*3%=0.09%
These are the probabilities for two slots
Do it one more time:
9%*30%=2.7%
0.36%*6%=0.0216%
0.09%*3%=0.0027%
So to get 3 commons it would ,on average, take 37 attempts
3 Normals would take 4629 attempts
3 rares would take 37037 attempts
You cube it because if we list all the probabilities off:
CCC
CCN
CCR
CNC
CNN
CNR
CRC
CRN
CRR
NCC
NCN
NCR
NNC
NNN
NNR
NRC
NRN
NRR
RCC
RCN
RCR
RNC
RNN
RNR
RRC
RRN
RRR
There are 27 different combinations there and three different possible options for each slot with repeats, 3cubed =27
If we couldn't have repeats (Forced to have one C, N, R per roll) then we would times it but because we can have repeats, we cube it.
Now read my signature because I spent 30 minutes writing this.