An analysis of a digital archive of ancient manuscripts reveals that 35% of the manuscripts are written in Latin. It is also found that 45\displaystyle \frac{4}{5}54 of the manuscripts in the archive contain hand-drawn illustrations. Of the manuscripts that contain hand-drawn illustrations, the probability that a randomly chosen manuscript is written in Latin is 0.3.
Find the probability that a manuscript chosen at random from the archive:
is written in Latin and contains hand-drawn illustrations.
is not written in Latin and does not contain hand-drawn illustrations.
Determine, giving your reason, whether or not the events 'written in Latin' and 'contains hand-drawn illustrations' are statistically independent.