# Independent/mutually excl. probability

Suppose a large # of students are surveyed about how they travel, $$G - 0.5$$, $$B - 0.4$$, $$W = 0.8$$

Given that: $$W$$ is independent of $$G$$ and $$W$$ is independent of $$B$$, but $$B$$ is mutually exclusive of $$G$$, what is the probability that a random student does none of them?

We want $$P(\overline{W} \cap \overline{G} \cap \overline{B})$$, but how can I split it?

• I think you mean $P[G]=0.5, P[B]=0.4, P[W]=0.8$. You might consider the identity $(W^c \cap B^c \cap G^c) \cup (W^c \cap B^c \cap G) = (W^c \cap B^c)$. – Michael Feb 2 '17 at 6:30

My previous answer was flawed. It is not generally true that if $W$ is independent of $G$ and $B$, then it is independent of $G \cap B$.
• @Michael, if $W$ is independent of $G$ and $W$ is independent of $B$, then why isnt it always true that $W$ is independent of $G \cap W$? – Aditya Kalra Feb 2 '17 at 6:38