You are told that:
- $P(P) = 0.15$
- $P(T|P) = 0.91$
- $P(T|¬P) = 0.04$
- $P$ and $T$ are not independent.
Whats $P(T)$ ?
I've been really struggling with this one. I've tried substituting a bunch of stuff into the multiplication rule $$P(A \cap B) = P(A) \cdot P(B|A)$$ and the addition rule $$P(A \cup B) = P(A) + P(B) − P(A \cap B).$$ I've also tried drawing a conditional probability tree to help me see what's going on, but I just don't seem to be getting anywhere.
Any level of help would be appreciated :)
$P(T)=P(T\cap P)+P(T\cap P^{c})=P(T|P)P(P)+P(T|P^{c}) P(P^{c})=P(T|P)P(P)+P(T|P^{c}) [1-P(P)]$.
[I have typed $A^{c}$ for the complement of $A$].