Security

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 k~1~, k~2~, ... , k~n~ 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^tk~1~^ ⋅ p^tk~2~^ ⋅ ... ⋅ p^tk~n~^ = 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:

1. they involve a fairly high mortality rate of 100%, and
2. they aren’t conspiracies in the first place.

Proof: The lemma holds with the following catch.

(P1) p^tk^ is still exponential over time t unless the sole conspirator, upon setting a plot in motion w.p. p^t~1~k^ = 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. ∎