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 

Twenty-sided dice roll 10.5 on average. Thirty-sided dice roll 15.5 on average. So Alice is going to roll 31.5 on average, while Bob only rolls 31 on average.
These are only averages, but the symmetry of the problem ensures Alice wins out (on average).
