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 ∪ B ∪ C) = 5P(A), then what is the value of P(C)?

Let P(A) = x. Then P(B) = 2x and P(C) = 4x. A and B disjoint means P(A∩B) = 0. A and C independent: P(A∩C) = P(A)P(C) = 4x². B and C independent: P(B∩C) = P(B)P(C) = 8x². P(A∩B∩C) ⊆ P(A∩B) = 0. P(A∪B∪C) = P(A) + P(B) + P(C) - P(A∩B) - P(A∩C) - P(B∩C) + P(A∩B∩C) = x + 2x + 4x - 0 - 4x² - 8x² + 0 = 7x - 12x². Setting this equal to 5P(A) = 5x: 7x - 12x² = 5x, so 2x = 12x², giving x = 1/6. Therefore P(C) = 4x = 4/6 = 2/3.