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I have this problem in my work. Assume I have multiple events, say A, B, C. I want to calculate probability P(ABC) But among A,B,C some of them may or may not be dependent to each other.

With regards to my work scenario, I have the values of probability P(A), P(B), P(C), and also probabilities like P(A|BC), P(B|AC),P(C|AB), P(A|C), P(B|C),P(A|B) etc. Since they are not independent, I can not use P(A,B,C)=P(A)*P(B)*P(C).

So I need to use conditional probability: P(ABC)=P(A|BC)*P(BC)=P(A|BC)*P(B|C)*P(C) Is this correct?

CyberPlayerOne
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  • Question is too vague. Is X = $A\cap B\cap C$ and what exactly is it you know? I suspect you need to look at Baye's Theorem, which will allow you to calculate a probability given enough known conditional probabilities. – Paul Apr 02 '14 at 08:50
  • Yes I think it is X = A∩B∩C in my scenario. – CyberPlayerOne Apr 02 '14 at 08:52

1 Answers1

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You always have that $$ P(X \cap Y) = P(X \mid Y)P(Y) = P(Y \mid X)P(X) $$ Applying that twice (first with $X=C,Y=A\cap B$, then with $X=B,Y=A$) yields $$ P(A \cap B \cap C) = P(C \mid A \cap B)P(A \cap B) = P(C \mid A \cap B)P(B \mid A)P(A) \text{.} $$ In plain english, this means that the probability that $A,B$ and $C$ occurs is the probability that $A$ occurs, times the probability that $B$ also occurs knowing that $A$ already occured, times the probability that $C$ occurs knowing that $A,B$ already occured.

fgp
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