The probabilities that a student passes in Mathematics, Physics and Chemistry…
2015
The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p and c respectively. Of these subjects, the student has 75% chance of passing in at least one, a 50% chance of passing in at least two and a 40% chance of passing in exactly two. Following relations are drawn in m, p and c:
(I) p + m + c = 27/20
(II) p + m + c = 13/20
(III) (p) × (m) × (c) = 1/10
- A.
Only relation 1 is true
- B.
Only relation 2 is true
- C.
Relations 2 and 3 are true
- D.
Relations 1 and 3 are true
Attempted by 48 students.
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Correct answer: D
Solution summary: apply the given probabilities and inclusion–exclusion for three events.
Given: P(at least one) = 3/4, P(at least two) = 1/2, P(exactly two) = 2/5.
Compute probability of passing all three: P(all three) = P(at least two) − P(exactly two) = 1/2 − 2/5 = 1/10.
Compute sum of pairwise intersections (S2). Using exactly-two = S2 − 3·P(all three), we get S2 = exactly-two + 3·P(all three) = 2/5 + 3·(1/10) = 7/10.
Apply inclusion–exclusion: m + p + c = P(at least one) + S2 − P(all three) = 3/4 + 7/10 − 1/10 = 27/20.
Conclusion: The relation giving m + p + c = 27/20 is true, the relation giving m + p + c = 13/20 is false, and the relation asserting the probability that the student passes all three is 1/10 is true. Therefore the correct pair of true relations is the first and the third.
Note: Relation three must be read as the probability of passing all three being 1/10 (not as the product m·p·c unless independence is stated).
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