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JEE Main 2014

JEE Main 2014 Mathematics question on Probability

Let AA and BB be two events such that P(A∪B‾)=16P(\overline{A \cup B}) = \frac{1}{6}, P(A∩B)=14P(A \cap B) = \frac{1}{4} and P(A‾)=14P(\overline{A}) = \frac{1}{4}, where A‾\overline{A} stands for the complement of the event AA. Then, the events AA and BB are
  1. A.
    independent but not equally likely
  2. B.
    independent and equally likely
  3. C.
    mutually exclusive and independent
  4. D.
    equally likely but not independent
Show answer and solution

Correct option: A

Solution

P(A) = 1 - 1/4 = 3/4. Since P(A∪B) = 1 - 1/6 = 5/6, we get P(B) = P(A∪B) - P(A) + P(A∩B) = 5/6 - 3/4 + 1/4 = 1/3. Now check independence: P(A)·P(B) = (3/4)·(1/3) = 1/4 = P(A∩B) ✓, so A and B are independent. Since P(A) = 3/4 ≠ P(B) = 1/3, they are not equally likely.

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