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IN A CLASS OF 130 STUDENTS.85 PASSED IN MATHS, 60 PASSED IN SCIENCE, 10 FAILED IN BOTH.

a) how many passed in science but failed in math
b)how many failed in exactly one subject

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2 answers

  1. If 10 failed in both, that means 130-10=120 passed at least one subject.

    Since 85 passed math, 60 passed science, there are 145 passes from 120 students. We conclude that 145-120=25 passed both.
    Hence (b) 120-25=95 passed (or failed) exactly one subject.

    For (a), 60 passed science, and 25 passed both, so 35 passed science but did not make math.

    A more mathematical treatment would be:
    P'(A∪B)=1-P(A∪B)
    P(A)=85/130
    P(B)=60/130
    P'(A∪B)=1-P(A∪B)=10/130
    =>
    P(A∪B)=(130-10)/130=120/130

    We also know that:
    P(A∪B)=P(A)+P(B)-P(A∩B)
    120/130=85/130+60/130-P(A∩B)
    =>
    P(A∩B)=85/130+60/130-120/130=25/130
    The cardinality of a set is the number of members in the set.
    For example, the cardinality of A is denoted by |A|, which is equal to the number of students who passed math=85.
    Hence |B|=60, |A∩B|=25, |A∪B|=120.
    (a)
    |A'∩B|=|B|-|A∩B|=60-25=35
    (b)
    |A∪B|-|A∩B|=120-25=95

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