The random variables $X$ and $Y$ are independent and have distributions $N(7, 3)$ and $N(6, 2)$ respectively. One value is chosen from each distribution. Find the probability that the absolute difference between the two chosen values is greater than $2$.
A candidate’s total score in a science test is worked out in the following way. Let $T$ stand for the theory mark and $P$ stand for the practical mark, and define the total score as $T + 1.5P$. Assume that $T$ and $P$ are independent and have distributions $N(62, 158)$ and $N(42, 108)$ respectively. You should take it that no continuity corrections are required when these distributions are used. A pass is awarded to candidates with a total score of at least $90$. Find the proportion of candidates who pass.
Comment on whether the variables $T$ and $P$ can reasonably be treated as independent.