Question

Classes A, B, and C are open to any of the 1000 majors in the department. The total enrollment figures in these classes are as follows: 150 majors in A, 180 in B, and 120 in C. There are 50 majors that are in both A and B, 25 that are in both A and C, and 30 that are in both B and C. There are 15 majors taking all three classes.

(a) If a student X is chosen randomly, with each of the 1000 majors equally likely, what is the probability that X is not in any of these three classes? Give an exact answer expressed as a simplified fraction.

(b) If a student X is chosen randomly, with each of the 1000 majors equally likely, what is the probability that X is in class B, given that X is in at least one of these three classes? Give an exact answer expressed as a simplified fraction.

(c) If two different majors are chosen randomly, with each of the pairs from the 1000 majors equally likely, what is the probability that at least one of this pair is taking none of these three classes? Give your answer to 3 significant digits.

Provide a brief 1-2 sentence explanation for each question.

Answer 1

total students in three classes =

n(A u B u C) = n(A)+n(B)+n(C)-n(A n B) - n(B n C) - n(A n C) + n(A n B n c)

=150+180+120-50-25-30+15=360

a)

total students that are not in any of these three classes = 1000 -360=640

P(probability that X is not in any of these three classes) = 640/1000 = 16/25

b) P(X from B | X is in  at least one of these three classes) = P(B)/P(A u B u C) = 180/360 = 1/2

c) P( at least one of this pair is taking none of these three classes) =P(that one has taken one of three classes and second has taken none of these) + P( both has taken none of these three classes) = 640C1*360C1/1000C2 +   640C2 / 1000C2 =

640*360/499500 + 204480/499500 = 0.871

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