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If we have probabilities for disjoint events:

$A, B, ..., \text{i.e.:}\space P(A), P(B), ..., \text{and}\space P(A) + P(B) + \ldots = 1$

then does this in fact mean, that there is a system, that has its activity (or in fact some abstract resources, that lead to the activity) partitioned between different tasks $A, B, \ldots$ ?

Seeing the probabilities as percentages of system's resources devoted to different tasks – is this a correct and useful approach, investigated in mathematics?

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Are the events supposed to be disjoint? –  Henry May 8 '12 at 6:33
    
If $A,B,\cdots$ etc are mutually exclusive events, then in what sense can a "system's resources" be distributed among them simultaneously? Unless you're into many-worlds interpretation type stuff. –  anon May 8 '12 at 6:34
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Your question is a little vague, to put it mildly. –  copper.hat May 8 '12 at 6:38

1 Answer 1

If there are two independent systems and the first allocates $p$ to task A, and the second allocates $q$ to task B, is there natural definition of "combined system" that allocates $pq$ to the combination of tasks A and B?

One possibility is to imagine "systems" as time-sharing mainframes that rapidly cycle between jobs, then looking at two different mainframes as a single computer whose jobs are pairs (job from system 1, job from system 2). But then it could be more complicated to talk about non-independent systems compared to the language used in probability theory.

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Event $A \cap B$ can mean temporally sequential occurrence of tasks A, B or B, A? –  Mooncer May 8 '12 at 8:13
    
If there is a time ordering the combination can be non-commutative, which is different from multiplication of probabilities. –  zyx May 8 '12 at 8:15
    
Then special case is for two events A, B, $P(A) = 1-P(B)$, because system is always in event A, or B, and having system then go through A, B or B, A is $P(A)P(B)$, like in math.stackexchange.com/questions/138692/… ? For three and more events the ordering would be important? –  Mooncer May 8 '12 at 8:27

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