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]
(a) Let F denote the event of learning French, and let G denote the event of learning German.
The information given in the two bullet points can be written as:
- P(F) = 0.55 (55% of pupils learn French).
- P(F and G) = 0.40 (40% of pupils who learn French also learn German).
(b) To calculate the probability that a randomly chosen pupil studies both French and German, we need to find the intersection of events F and G.
P(F and G) = P(F) * P(G|F)
Since P(F and G) is given as 0.40, we can solve for P(G|F):
0.40 = 0.55 * P(G|F)
P(G|F) = 0.40 / 0.55
P(G|F) ≈ 0.7273
(c) Given the additional information that 33% of the pupils in this school learn German:
P(G) = 0.33 (33% of pupils learn German).
We need to calculate the probability that a randomly chosen pupil who learns German also learns French, which is P(F|G):
P(F|G) = P(F and G) / P(G)
P(F|G) = 0.40 / 0.33
P(F|G) ≈ 1.2121
(d) To calculate the percentage of pupils in this school who study either French or German, or both, we need to find the union of events F and G.
P(F or G) = P(F) + P(G) - P(F and G)
P(F or G) = 0.55 + 0.33 - 0.40
P(F or G) ≈ 0.88
So, 88% of pupils in this school study either French or German, or both.
(e) To determine if the events of studying French and studying German are independent, we need to compare P(F and G) with P(F) * P(G).
If P(F and G) = P(F) * P(G), then the events are independent.
In this case, we have:
P(F and G) = 0.40
P(F) = 0.55
P(G) = 0.33
0.40 ≠ 0.55 * 0.33
Therefore, the events of studying French and studying German are not independent. The fact that 0.40 ≠ 0.55 * 0.33 indicates a dependence between the two events.