maths probability help

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.

(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.
(d) What percentage of pupils in this school study either French or
German, or both?
(e) For pupils in this school, are the events of studying French and
studying German independent? Give a reason for your answer.
Please help

Answer 1

From the initial information:

 

Of the 55% that learn French, we have a crossover

of 40% of that 55% also learn German.

 

.55(.4) = .220 = 22%

 

Thus, of the French learning students we

have 33% learn only French, 22% learn French and German.

 

(c) we know 33% of the students learn German.

We already know that 22% learn French and German.

This means that 11% of the students learn only German.

 

Thus we have twice as many students learning German plus

French as we have learning only German.

 

The probability that a randomly chosen student who learns

German also learns French is (G+F):G = 22:11 = 2:1

 

d)  The total percentage of students that learn either

French or German or both is:

 

F + (F+G) + G = 33% + 22% + 11% = 66%

 

e)  66% of the students in the school study either

French, German, or both.  Since there is a crossover

of 22% that study both, then overall the events of

studying French and studying German are not

independent.