Probability of picking more balls of one color Suppose you flip a 3-sided coin $n$ times. The sides are denoted: $A$, $B$, and $C$. The probability of a coin flip turning one of the sides is given by $p_A$, $p_B$, and $p_C$, respectively. 
What is the probability that you end up flipping side $A$ more times than side $B$?

I know this looks like a homework problem, but it is not. It is a rewording of a problem I ran into in my research. It looks easy, but I am having trouble getting it started. Any hints would be very welcome. 

The probability is $1$ if p(A) > p(B) otherwise it is $0$.
EDIT
My answer above was based on the first version of this question.
