It is actually quite easy to get in CS master program if you don’t mind state schools. I have a degree in Structural Engineering at UCSD. I don’t like what I am doing right now and I really enjoy programming. Thus I applied to CSU Fullerton, Long Beach, Pomona, and LA for fall 2013 and got accepted to all except Long Beach. I only need to take 4 extra CS courses since I have an extensive background in Math and a bit of CS.
My friend is also in the same situation as you. He got in Fullerton and LA. He had a degree in Psychology at SDSU. However, he has to take at least 8 additional courses (5 CS and 3 Math).
I recommend you to take at least a Java/C++ class at community college to see whether if you like it or not.