BinomPDF[k_, N_, dt_, f_] := 
      PDF[BinomialDistribution[N, 1 - Exp[-f*dt]], k]

Trigger1[nTh_, N_, dt_, f_] := 
      Evaluate[Sum[
            BinomPDF[k, N, 5/10^9, f]*
              BinomPDF[nTh - k, N, dt, 
                f]*(1 - Sum[BinomPDF[j, N, 5/10^9, f], {j, 0, k}]), {k, 
              0, nTh}]]*(10^9/5)

?BinomPDF

?Trigger1

Trigger1[nTh_, N_, dt_, f_] := 
      Evaluate[Sum[
            BinomPDF[k, N, 0.0001/10^9, f]*
              BinomPDF[nTh - k, N, dt, 
                f]*(1 - 
                  Sum[BinomPDF[j, N, 0.0001/10^9, f], {j, 0, k}]), {k, 0, 
              nTh}]]*(10^9/0.0001)

?Trigger1

N[Trigger1[249, 81000, 425/10^9, 6 10^3]]