There are 400 students in a class. Of these, 270 study Maths, 300 French, and 50 Business Studies. All those who study Maths also study French. 20 study Maths and Business Studies. 35 study French and Business Studies. A student is selected at random. Find the probability that the selected student studies French but neither Maths nor Business Studies. Use a Venn diagram only.

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Points to Remember:

  • Understanding of set theory and probability.
  • Application of Venn diagrams to solve probability problems.
  • Accurate calculation of probabilities.

Introduction:

This question involves calculating conditional probability using a Venn diagram. We are given the number of students studying different subjects (Maths, French, Business Studies) in a class of 400 students, along with information about the overlap between these subjects. The goal is to determine the probability that a randomly selected student studies French but neither Maths nor Business Studies. This requires visualizing the student population using a Venn diagram and calculating the relevant probabilities.

Body:

1. Constructing the Venn Diagram:

Let’s represent the number of students studying each subject using a Venn diagram. We’ll use the following notation:

  • M: Students studying Maths
  • F: Students studying French
  • B: Students studying Business Studies

We are given:

  • Total students (T) = 400
  • |M| = 270
  • |F| = 300
  • |B| = 50
  • All students studying Maths also study French (M ⊂ F)
  • |M ∩ B| = 20
  • |F ∩ B| = 35

Since all students studying Maths also study French, the intersection of M and F is simply the number of students studying Maths, i.e., |M ∩ F| = 270.

Now, we can fill in the Venn diagram:

[Unfortunately, I can’t create visual diagrams here. Imagine a Venn diagram with three overlapping circles representing M, F, and B. The following steps describe how to populate it.]

  • Start with the intersection of all three sets: Since |M ∩ B| = 20 and M is a subset of F, then |M ∩ F ∩ B| = 20. Write ’20’ in the area where all three circles overlap.

  • Next, find the number of students studying only French and Business Studies: |F ∩ B| – |M ∩ F ∩ B| = 35 – 20 = 15. Write ’15’ in the area where only F and B overlap.

  • Find the number of students studying only Maths and Business Studies: This is 0 because all students studying Maths also study French.

  • Find the number of students studying only Maths: This is also 0 because all students studying Maths also study French.

  • Find the number of students studying only French: This is |F| – |M ∩ F| – |F ∩ B| + |M ∩ F ∩ B| = 300 – 270 – 15 + 20 = 35. Write ’35’ in the area representing only French.

  • Find the number of students studying only Business Studies: |B| – |M ∩ B| – |F ∩ B| + |M ∩ F ∩ B| = 50 – 20 – 15 + 20 = 35. Write ’35’ in the area representing only Business Studies.

2. Calculating the Probability:

The number of students studying French but neither Maths nor Business Studies is the number of students in the region representing only French, which we calculated as 35.

Therefore, the probability that a randomly selected student studies French but neither Maths nor Business Studies is:

P(F only) = (Number of students studying only French) / (Total number of students) = 35/400 = 7/80

Conclusion:

The probability that a randomly selected student studies French but neither Maths nor Business Studies is 7/80 or 0.0875. This calculation was achieved through the systematic construction and interpretation of a Venn diagram, allowing for a clear visualization of the overlapping sets of students studying different subjects. The use of a Venn diagram provides a straightforward and effective method for solving probability problems involving multiple intersecting sets. This approach emphasizes the importance of visual aids in understanding and solving complex probability questions. Further analysis could involve exploring the correlation between subject choices and student demographics or academic performance.

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