Question

In a class of 50 students it is known that: 35 study in English course, 15 study both in English and French courses, 25 study in French course, 13 study both in French and German courses, 28 study in German course. a) What is the maximum and minimum of the probability of a randomly chosen student from this class to study in all three language courses? b) What is the maximum of the probability of a randomly chosen student from this class not to study in any of these language courses?

Answer 1

N(A ∪ B ∪ C) = N(A) + N(B) + N(C) − N(A ∩ B) − N(A ∩ C) − N(B ∩ C) + N(A ∩ B ∩ C)

here,

A = english

B = french

C = german

a.

since N(A ∩ B) + N(B ∩ C) = N(B) + 1

so, minimum N(A ∩ B ∩ C) = 1

and since the smallest 2 set intersection is N(B∩C) = 13

so, N(A ∩ B ∩ C) can be maximum 13

minimum probability = 1/50 = 0.02

max. probability = 13/50 = 0.26

b.

maximum N(none) = 50 - minimum of N(A ∪ B ∪ C)

for minimum N(A ∪ B ∪ C) :

so we get minimum N(A ∪ B ∪ C) = 5+2+13+15+10 = 45

maximum N(none) = 50 - minimum of N(A ∪ B ∪ C) = 50-45

= 5

maximum of the probability of a randomly chosen student from this class not to study in any of these language courses = 5/50 = 0.1

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