Your colleagues in a lab have sequenced DNA from a large population in order to understand how a gene (G) influences two particular traits ($T_1$ and $T_2$). They find that $P(G)=0.6$, $P(T_1|G)=0.7$, and $P(T_2|G)=0.9$. They also observe that if a subject does not have the gene $G$, they express neither $T_1$ nor $T_2\space$ [$P(T_1|\overline G)=P(T_2|\overline G)=P(T_1\cap T_2|\overline G)=0$]. The probability of a patient having both $T_1$ and $T_2$ given that they have the gene $G$ is $0.63$.
In the case above, I want to see if $T_1$ and $T_2$ are independent
My work: In order to check independence of those two events, I want to see if $(_1\cap _2)=(_1)(_2)$. If this equation occurs, then $T_1$ and $T_2$ would be independent. I am stuck at trying to extract the "$G$" from $P(T_1 | G) = 0.7$, and $P(T_2 | G) = 0.9$ which are information given in the problem. How can I do this so that I can get $P(T_1)$ and $P(T_2)$? Also, how can I extract the $G$ from $(_1\cap _2|G) = 0.63$ so that I can find $(_1∩_2)$.
Thanks in advance!
As per the law of total probability $$P(T_1)=P(G).P(T_1|G)+P(\overline G).P(T_1|\overline G)=P(G).P(T_1|G)=0.7P(G)$$ This is because as per your question, $P(T_1|\overline G)=0$
Similarily $$P(T_2)=0.9P(G)$$ Now $$P(T_1\cap T_2)=P(G).P(T_1\cap T_2|G)+P(\overline G).P(T_1\cap T_2|\overline G)=P(G).P(T_1\cap T_2|G)=0.63P(G)$$ Now for $T_1$ and $T_2$ to be independent, $P(T_1\cap T_2)=P(T_1).P(T_2)$ $$P(T_1\cap T_2)=0.63P(G)=0.378$$ $$P(T_1).P(T_2)=0.63[P(G)]^2=0.2268$$ As we can see, $P(T_1\cap T_2)\neq P(T_1).P(T_2)$
$\therefore T_1$ and $T_2$ are not independent.