True, and interesting since this can be used as a statistical lever to ignore the exponential scaling effect of conditional probability, with a minor catch.
Lemma:
Compartmentalization can reduce, even eliminate, chance of exposure introduced by conspirators.
Proof:
First, we fix a mean probability p of success (avoiding accidental/deliberate exposure) by any privy to the plot.
Next, we fix some frequency k1, k2, ... , kn of potential exposure events by each conspirators 1, ..., n over time t and express the mean frequency as k.
Then for n conspirators we can express the overall probability of success as
1 ⋅ p^tk1^ ⋅ p^tk2^ ⋅ ... ⋅ p^tkn^ = p^ntk^
Full compartmentalization reduces n to 1, leaving us with a function of time only p^tk^. ∎
Theorem:
While it is possible that there exist past or present conspiracies w.h.p. of never being exposed:
they involve a fairly high mortality rate of 100%, and
they aren’t conspiracies in the first place.
Proof:
The lemma holds with the following catch.
(P1) p^tk^ is still exponential over time tunless the sole conspirator, upon setting a plot in motion w.p. p^t1k^ = p^k^, is eliminated from the function such that p^k^ is the final (constant) probability.
(P2) For n = 1, this is really more a plot by an individual rather than a proper “conspiracy,” since no individual conspires with another. ∎