Betting game: $3$ $20$-sided dice versus $2$ $30$-sided dice 
Alice and Bob play a dice game. Alice rolls three $20$-sided dice and
  Bob rolls two $30$-sided dice. Whoever has the highest sum wins.
Who has a better chance of winning?

This was a question asked in an interview. So I don't have much time to answer and I want to find the answer quickly. I did it the long way (enumerate all possibilities), and I figured out that Alice has a better chance of winning. But is there any way I can solve this problem very quickly?
I tried to make a "flipping over the dice" argument but got nowhere.
Thanks 

The flipping dice argument is straight forward. When flipping a dice you have that $n$ turns into $21-n$ (or $31-n$ depending on which type of die).
What you do is note that if Alice win for some outcome Bob will win in the equally probable(*) outcome when all dice are flipped. This is because if Alice rolled $a$ and Bob $b$ in total and $a>b$ the result with dice flipped is Alice rolling $21-a$ and Bob $31-b$ and because $a>b$ we have $21-a < 21-b < 31-b$.
The outcomes in the preceding paragraph is as many wins for Bob as it's far Alice. For outcomes that are not covered by the preceding paragraph there is no win for Alice. If we just can find a single win for Bob amongst these there is more outcomes where Bob wins and he would be more likely to win than Alice. But there is, consider for example Alice rolling $21=11+10$ and Bob rolling $47=16+15+16$ which Bob wins and he wins even if the dice are flipped (Alice rolling $21=10+11$ and Bob $46=15+16+15$) so the case is not covered by the preceding paragraph.
(*) All outcomes will be equally probable if we consider each die uniquely identified (ie distingishing Alice rolling $10+11$ from Alice rolling $11+10$).
