• Septimaeus@infosec.pub
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    7 months ago

    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 ⋅ ptk~1~ ⋅ ptk~2~ ⋅ … ⋅ ptk~n~ = pntk

    Full compartmentalization reduces n to 1, leaving us with a function of time only ptk. ∎

    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) ptk is still exponential over time t unless the sole conspirator, upon setting a plot in motion w.p. pt~1~k = pk, is eliminated from the function such that pk 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. ∎