<< Statistics`DiscreteDistributions`

PoissPDF[k_, N_, dt_, f_] := 
      PDF[PoissonDistribution[N*f*dt], k]

Trigger0[nTh_, N_, dt_, f_] := 
      Evaluate[Sum[
            PoissPDF[k, N, 5/10^9, f]*
              PoissPDF[nTh - k, N, dt, 
                f]*(1 - Sum[PoissPDF[j, N, 5/10^9, f], {j, 0, k}]), {k, 
              0, nTh}]]*(10^9/5)

Trigger[nTh_, N_, dt_, f_] := 
      Evaluate[Sum[
            PoissPDF[k, N, 0.0001/10^9, f]*
              PoissPDF[nm - k, N, dt - 0.0001/10^9, 
                f]*(1 - 
                  Sum[PoissPDF[j, N, 0.0001/10^9, f], {j, 0, 
                      k + nTh - nm}]), {nm, 0, nTh}, {k, 0, nm}]]*(10^9/
            0.0001)

?Trigger

N[Trigger0[52, 81000, 425/10^9, 10^3]]

N[Trigger[52, 81000, 425/10^9, 1 10^3]]