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probability question
Posted:
Jul 6, 1996 11:52 PM
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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.
Al Aburto
aburto@cts.com
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