I am preparing for midterm exam and need to know what is the step by step solution to this question? Answer is shown in red. Also any external related link is very much appreciated.
First, there is a standard factorization of probability that always holds:
$$P(\sim A , \sim B , \sim C) = P(\sim A) P(\sim B | \sim A)P(\sim C | \sim B, \sim A) $$
Given the shape of the Bayesian network, we conclude that $C$ is independent of $A$ given $B$. That means:
$P(\sim C | \sim B, \sim A) = P(\sim C | \sim B)$
Since we know $P(A) = 0.3$, we also know $P(\sim A) = 0.7$. Furthermore $P(\sim B | \sim A) = 1 - P(B | \sim A) = 1 - 0.5 = 0.5$. Finally $P(\sim C | \sim B) = 1 - P(C | \sim B) = 1 - 0.1 = 0.9$
Putting it all together we get $$ P(\sim A , \sim B \sim C) = 0.7 \cdot 0.5 \cdot 0.9 = 0.315$$