transmissionProcess=BinomialProcess[10^-4];

prob=Probability[x>0,x\[Distributed]transmissionProcess[n]]

ListLogPlot[Transpose@Table[{prob,10^(-3)},{n,0,15}],Joined->True,Filling->Axis]

Maximize[{n,prob<=10^-3&&n∈Integers&&5<=n<=15},n]