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
To solve these questions, we can use conditional probability and the formula for calculating the probability of the union of two independent events.
(a) The probability that a randomly chosen pupil in this school who learns German also learns French can be calculated using conditional probability. Let's denote the events as follows:
A = learning French
B = learning German
We are given:
P(A) = 0.55 (55% choose to learn French)
P(A|B) = 0.40 (40% of French learners also learn German)
The probability that a randomly chosen pupil who learns German also learns French can be calculated as:
P(A and B) = P(A|B) * P(B)
P(A and B) = 0.40 * 0.33 (33% learn German)
Therefore, P(A|B) = P(A and B) / P(B) = (0.40 * 0.33) / 0.33 = 0.40
So the probability that a randomly chosen pupil who learns German also learns French is 0.40 (or 40%).
(b) To find the percentage of pupils in the school who study either French or German, or both, we can add the probabilities of studying French and studying German and then subtract the probability of studying both:
P(A or B) = P(A) + P(B) - P(A and B)
We are given:
P(A) = 0.55 (55% choose to learn French)
P(B) = 0.33 (33% learn German)
P(A and B) = P(A|B) * P(B) = 0.40 * 0.33 = 0.132
P(A or B) = 0.55 + 0.33 - 0.132 = 0.748
So the percentage of pupils in the school who study either French or German, or both, is 74.8%.
(c) To determine whether studying French and studying German are independent events, we need to compare the conditional probability P(A|B) with the marginal probability P(A).
If P(A|B) = P(A), then the events are independent. Otherwise, they are dependent.
From part (a), we found that P(A|B) = 0.40 and P(A) = 0.55.
Since P(A|B) ≠ P(A), we can conclude that studying French and studying German are dependent events.
The reason for this dependence is that the probability of studying French varies depending on whether a pupil is already learning German or not.
(c) To calculate the probability that a randomly chosen pupil in this school who learns German also learns French, you can use the concept of conditional probability.
Let's denote the event "choosing French" as F and the event "choosing German" as G. We want to find P(F|G), which represents the probability of choosing French given that German is chosen.
We are given that 55% of pupils choose to learn French, so P(F) = 0.55.
We are also given that of the pupils who learn French, 40% also learn German, so P(G|F) = 0.40.
Using the formula for conditional probability, P(F|G) = P(F ∩ G) / P(G), we need to find P(F ∩ G).
Since P(G) is not directly given, we can find it using the fact that 33% of the pupils learn German. So P(G) = 0.33.
Now, we can calculate P(F ∩ G) as P(F ∩ G) = P(G|F) * P(F) = 0.40 * 0.55 = 0.22.
Finally, we can determine P(F|G) using the formula: P(F|G) = P(F ∩ G) / P(G) = 0.22 / 0.33 ≈ 0.67.
Therefore, the probability that a randomly chosen pupil in this school who learns German also learns French is approximately 0.67, or 67%.
(d) To find the percentage of pupils in this school studying either French or German, or both, we can use the concept of union and the formula for probability of the union of two events.
Let's denote the event "choosing French" as F and the event "choosing German" as G. We want to find P(F ∪ G), which represents the probability of choosing French or German, or both.
We know that P(F) = 0.55 (55% choose French) and P(G) = 0.33 (33% choose German). However, we also need to account for the overlap between the two events, as some pupils choose both French and German.
To find P(F ∪ G), we can use the formula: P(F ∪ G) = P(F) + P(G) - P(F ∩ G).
From part (c), we found that P(F ∩ G) = 0.22 (22% choose both French and German).
So, P(F ∪ G) = 0.55 + 0.33 - 0.22 = 0.66.
Therefore, the percentage of pupils in this school studying either French or German, or both, is 66%.
(e) To determine whether the events of studying French and studying German are independent for the pupils in this school, we need to check if the probability of one event is affected by the occurrence of the other event.
If P(F) = P(F|G) or P(G) = P(G|F), then the events are independent. But if these conditions are not met, the events are dependent.
From part (c), we found that P(F|G) ≈ 0.67, whereas P(F) = 0.55 and P(G) = 0.33.
Since P(F|G) ≠ P(F) and P(G) ≠ P(G|F), the events of studying French and studying German are dependent for the pupils in this school. The probability of studying French is affected by whether or not German is chosen, and vice versa.
Therefore, the events of studying French and studying German are not independent for the pupils in this school.