Hi,

I am unsure whether my arguments below are correct.

Suppose that P(A, B, C) is a discrete joint distribution. My goal is to express P(A | B) in terms of P(A | B, C).

My solution:

$\displaystyle P(A | B) = \dfrac{P(A \cap B)}{P(B)}$

$\displaystyle P(A | B) = \sum_c \dfrac{P(A \cap B \cap \{c\})}{P(B)} = \sum_c \dfrac{P(A, B |\{c\})}{P(B)}P(\{c\}) = \sum_c P(A | B, \{c\})P(\{c\})$

Thanks and best regards,

samosa