P(A ^ B ^ C) given independence of B and C If you are given P(B^C) = P(B)P(C) (so B and C are independent), and you know P(A^C), P(A^B), P(A), P(B), and P(C), but A and B are dependent, and A and C are dependent, is it possible to calculate P(A^B^C)?
Since you know that B and C are independent, can you say that (A^B) and (A^C) are independent? It doesn't feel like you can, but that's the only way I can think of to solve the problem.

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*$B$ the result of the green die is 2 or 3

*$C$ the result of the red die is 1, or 2 

*$A$ the sum of the two dice is 2, 3, or 5
$$\begin{array}{ll:ll} A\cap B &= \{(2,1), (2,3), (3,2)\} & P(A\cap B) & =1/12\\A\cap C & = \{(1,1), (1,2), (2,1)\} & P(A\cap C) & =1/12\\A\cap B\cap C & = \{(2,1)\} & P(A\cap B\cap C) & = 1/36\end{array}$$


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*Witness $B$ and $C$ are independent events, but $A\cap B$ and $A\cap C$ are not independent events.



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