# Given P(A|B) and P(B|C), calculate P(A|C)

My question concerns conditional probability and chaining those distributions.

I have not been able to find a rule that allows me to derive $$P(A|C)$$, when $$P(A|B)$$ and $$P(B|C)$$ are given for three dependent events A, B, and C.

I believe it is not true that $$P(A|C) = P(A|B) * P(B|C)$$. Is it possible to calculate $$P(A|C)$$, if more information is known, such as $$P(B)$$ or $$P(C)$$

Any pointers or references are welcome.

• The events are independent? Then $P(A\,|\,B)=P(A)$ and so on.
– lulu
Commented Apr 19, 2019 at 14:22
• and if they are dependent?
– Amir
Commented Apr 19, 2019 at 14:23
• You can say $P(A \mid C)= P(A \mid B, C)P(B \mid C)+ P(A \mid B^c, C)P(B^c \mid C)$ Commented Apr 19, 2019 at 14:23
• If $A$'s relationship with $C$ just depended on $B$, i.e. $P(A \mid B, C)=P(A \mid B)$ and $P(A \mid B^c, C)=P(A \mid B^c)$ then you could also say $P(A \mid C)= P(A \mid B)P(B \mid C)+ P(A \mid B^c)P(B^c \mid C)$; if in addition $A \subset B$ so $P(A \mid B^c)=0$ then you could say $P(A \mid C)= P(A \mid B)P(B \mid C)$. But this requires some strong assumptions which generally are not correct Commented Apr 19, 2019 at 14:29