First off: I am not familiar with the details of quantum mechanics or relativity. My understanding of Hawking radiation is as follows: Pairs of particles and anti-particles can and will spontaneously form in space and when that happens near the event horizon of a black hole the anti-particle might fall into the black hole and the particle not, effectively causing the black hole to lose mass.
Now, this is totally fine with me. The thing I do not understand is: Why do more anti-particles fall into the black hole than particles? If these pairs pop up randomly, shouldn't the effects of Hawking-radiation and anti-Hawking-radiation cancel out?