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I have P(A | B, C). I know P(B | C), P(B) and P(C). I need to figure out P (B | C, A). I am trying to use the chain rule and still unable to get to where I need to. Any clues would be useful.

P(B|C,A) = P(B,C,A)/P(C,A). I can find P(C,B,A) but how do I find P(C,A)?

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We have the following relationship, and you have all the things in the numerator part: $$P(B|C,A)=\frac{P(A|B,C)P(B|C)}{P(A|C)}$$

But, you don't know $P(A|C)$ or anything else that will help you to calculate it.

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  • $\begingroup$ Thanks @gunes: So, some information is missing in the problem is what you are suggesting? $\endgroup$
    – user1408865
    Commented Dec 18, 2020 at 19:35
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    $\begingroup$ Yes, that's correct @user1408865 $\endgroup$
    – gunes
    Commented Dec 18, 2020 at 19:39
  • $\begingroup$ I re-read the question. I do have P(A | B). I was able to get P(C | B, A). But unable to get P (B | C, A). Can you help look now? Thanks. $\endgroup$
    – user1408865
    Commented Dec 18, 2020 at 20:35
  • $\begingroup$ Still, it doesn't seem possible. Can you share the source of the question? $\endgroup$
    – gunes
    Commented Dec 18, 2020 at 21:01

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