upon the message detection probability, PD, is obtained by averaging (1.12) over the probability distribution of (1.11), namely, (1.13) Equation (1.13) cannot be easily calculated without the use of a computer; Low Probability of Intercept Communications however, with N large and PI very small, the approximation PD III PD(n) (1.14) is valid, where n is the mean of the distribution in (1.11), namely, (1.15) Substituting (1.15) into (1.12) and ignoring the (1 - P- factor gives (1.16) Solving for PI yields the desired result = 1 - (1.17) The companion relationship for PI is = PFA/(WssTMf). (1.18) Equations (1.17) and (1.18) are therefore used in lieu of (1.8a) and (1.8b), respectively, for computing the performance of the PB-FBC as per (1.9). The preceding gives the average or expected performance of the PB-FBC for a single message having Nh hops. It is instructive to examine the condi- tional detection probability in order to gain an understanding of the true sta- tistical nature of per-message detection using a PB-FBC. For this purpose, it is necessary to return to the numerical example. A recapitulation of the parameter values is: PD=0.9 PFA==10-6 = NT = 106 = Nh=8000. Suppose that it is decided that an N'T = 1000-channel PB-FBC is practical (rather than the one-million-channel FBC), then f = Phit = N'T/NT = 10-3. The per-channel probabilities calculated using (1.17) and (1.18) become P1=0.25 and PI = 1.25 X 10-13 (Contrast these numbers with PDI = 1.25 X 10-4 andPFAI=1.25 x 10-16 fr the one-million-channel FBC). Also note that n = Nhf = 8, i.e., this is the average number of hits per message. Table 1.1 lists P(N), PD(N), and the cumulative probability, PR {n < N} as functions of N. The interpretation is that, for any given message of 8000 hops, there could be N hits with probability P(N) and the corresponding condi- Interception Detectors Table 1.1 Binomial distribution detection probabilities, PD = PD(n) = 0.000334 0.000334 0.002675 0.250 0.003009 0.010712 0.438 0.013722 0.028587 0.578 0.042308 0.057209 0.684 0