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Does changing two events, $A$ and $B$ for example, to mutually exclusive in a new/altered situation, change their probability from the precondition? Would the $P(A)$, probability of $A$, now equal the $P(A\text{ and ~}B)$, probability of $A$ and not $B$?

Specific Example:

Precondition: A house is being inspected, and it may fail for any of the following reasons, each with probabilities as given: radon - 0.01 mould - 0.02 CO - 0.03 rodents - 0.04 Assume that the various hazards/infestations can occur simultaneously and independently.

New Situation: Suppose that you have learned that the inspection was failed, but you have not yet been told the reason.
Additionally, suppose that CO and rodents are mutually exclusive. What is the probability that the house had a rodent infestation?

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Can't be done, not enough information. – André Nicolas Sep 26 '12 at 0:39
what is your prior belief about the joint distribution over the 16 ways to be infested or not infested with radon mould CO and rodents, other than knowing the marginal probabilities – binn Sep 26 '12 at 0:51
We can assume that the various hazards/infestations can occur simultaneously and independently. – user937897 Sep 26 '12 at 1:11
up vote 0 down vote accepted

Let's compare the cases where the problems are independent to where they are all exclusive. If they are all exclusive, the chance that a failed house has rodents is $\frac 4{10}$ and the chance that a house fails at all is $\frac 1{10}$. If they are independent, the chance that a house fails is $1-0.99\cdot 0.98 \cdot 0.97 \cdot 0.96=0.09654976$. The chance that a failing house has rodents is then $\frac {0.04}{0.09654976}\approx 0.4142941422$. It seems reasonable that the probability has gone up some because some houses have more than one problem.

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