Three events A, B and C are such that A and B are disjoint, A and C are independent, B and C are independent. If 4P(A) = 2P(B) = P(C) and P(A \cup B \cup C) = 5P(A), then what is the value of P(C) ?
- A. \frac{5}{6}
- B. \frac{1}{3}
- C. \frac{1}{6}
- D. \frac{2}{3} ✓
Correct Answer: D. \frac{2}{3}
Explanation
Let P(A)=x, P(B)=2x, P(C)=4x. Substituting into the inclusion-exclusion formula: 5x = x + 2x + 4x - 0 - x(4x) - 2x(4x) + 0. This simplifies to 5x = 7x - 12x^2, yielding 12x^2 = 2x, so x = 1/6. Thus, P(C) = 4x = 2/3.
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