The key is to make yourself stand out from the rest, then your odds will start shifting.
The odds when I went in for my interviews for my current position was 500+ to 1, where I didn't go to school for the field I applied for. Sure, a bit of luck was involved, but being different from the other applicants is what makes you stand out of the hundreds seen. When speaking to HR the following year, they say it was mostly my personality and the way that I presented myself that got me the job (alongside a bit of experience).
Generally speaking (obviously depending on the job), they want to hire people that they can work with and can handle their responsibilities.
After going to a few interviews, I gained confidence in how I spoke and presented myself as an easy going open person who took care of his responsibilities that could carry out a conversation. The worst thing you can do in an interview to blend in with others and simply answer Yes/No to every question. If that is all you have to say, they could have just sent you a e-mail and said "fill this out". You basically write yourself off..
Admittedly, that's how I handled the first few interviews which didn't turn out well which is why I self-evaluated myself in what I was doing wrong.