Probabilities of studying exactly two, only one and none of three languages from a three-set Venn diagram

Probabilities of studying exactly two, only one and none of three languages from a three-set Venn diagram

6 marks

A school asked a group of \(120\) of its students which of three languages they learn: Chinese \((C)\), Japanese \((J)\) and Spanish \((S)\). In total, \(40\) of the students learn Chinese, \(35\) learn Japanese and \(30\) learn Spanish. Some of the numbers of students are given in the Venn diagram below.

One of the \(120\) students is selected at random. Find the probability that the selected student

  1. learns exactly two of the three languages;[1]
  2. learns Japanese but neither of the other two languages;[2]
  3. learns none of the three languages.[3]

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Answer:(a) \(\frac{19}{120}\) (b) \(\frac{1}{8}\) (c) \(\frac{11}{30}\)

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