Question

A large school in the UK offers its pupils the opportunity to learn several
non-English languages. It turns out that:
• 55% of pupils in this school choose to learn French;
• of the pupils who learn French, 40% also learn German.
(a) For pupils in this school, let F denote the event of learning French,
and let G denote the event of learning German. Write the information
given in the two bullet points in symbolic form. [2]
(b) Calculate the probability that a randomly chosen pupil in this school
studies both French and German. [3]
(c) Additional information is now given that 33% of pupils in this school
learn German. Calculate the probability that a randomly chosen pupil
in this school who learns German also learns French. [3]
(d) What percentage of pupils in this school study either French or
German, or both? [3]
(e) For pupils in this school, are the events of studying French and
studying German independent? Give a reason for your answer. [2]