I'm trying to come to grips with the following probability problem, but I'm unsure how figure out the logic and find the answer. So I'll pose the problem, question, and plea for help here.
Problem: A signal is transmitted across the Galaxy from and alien civilization. A receiver across the Galaxy (100,000 Light Years away) has the right equipment to detect the signal. Assume for the moment that the receiver is looking at the right place in the sky and the right frequency band to detect the transmitted signal withou error. The only problem on the receiver side is they do not know when the signal was transmitted during the lifetime of the Galaxy and its stars.
If one assumes 10 billion years (Tg) for the lifetime of the Galaxy then the probability of detection (Pd) of this one event is Pd = Tl / Tg, where Tl is how long one looks for the signal at the reciever.
This is fine, but my problem is: Suppose now there are N civilizations transmitting uniformly randomly over time. What then is the probability of detection of one of those events?
It cannot be PD = Pd^(N) (Pd to the N'th power) because the new total probability wound decrease as N increased which makes no logical sense. I thought it could be PD = N * Pd, but this can not be since if N were very large the new probability would no longer be normalized to one. Well, I'm at a loss to find, reason through, the solution rigorously.