Success = p failure = (1-p) is the correct first step.
To find the probability of at least one success, P(X>0), it is much easier to first calculate the probability of P(X=0)=NO success. Then the complement of P(X=0)'=1-P(X=0) = P(>0) = probability of at least one success.
By the multiplication rule, P(X=0)=(1-p)^9 or the binomial distribution P(X=0)=C(9,0)(p^0)(1-p)^9=1*1*(1-p)^9=(1-p)^9
So the probability of at least one success = 1-P(X=0) =1-(1-p)^9
