There are $\binom{12}{3} = 220$ ways to choose three numbers from the white balls, and $\binom{20}{1} = 20$ ways to choose one number from the red balls. Hence, the total number of possible outcomes when you buy a ticket is $220 \cdot 20 = 4400.$
For you to win the super prize, you need either:
Two of your white balls to match the three white balls drawn, which can be achieved in $\binom{3}{2} = 3$ ways, or
One of your white balls to match one of the three white balls drawn, and your red ball to match the red ball drawn, which can be achieved in $3 \cdot 1 = 3$ ways.
All three of your white balls to match the three white balls drawn, which can be achieved in $\binom{3}{3} = 1$ way.
So the number of favorable outcomes is $3 \cdot \binom{3}{2} + 3 \cdot 1 + 1 = 10.$
The probability of winning a super prize is 10/4400 = 1/440.
( If you want to verify , verify it by yourself). Its just so easy!