If events A, B and C are independent events and P(A) = P(B) = P(C) = p

1.	If events A, B and C are independent events and P(A) = P(B) = P(C) = p, then find the value of P(A ∪ B ∪ C) in terms of p.
Answer» Events A, B and C are independent events. P(A) = P(B) = P(C) = p P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P (A ∩ B) – P (A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C) = P(A) + P(B) + P(C) – P(A) ∙ P(B) – P(A) ∙ P(C) – P(B) ∙ P(C) + P(A) ∙ P(B) ∙ P(C) = p + p + p – (p × p) – (p × p) – (p × p) + (p × p × p) = 3p – p² – p² – p² + p³ = 3p – 3p² + p³ = p (3 – 3p + p²)

If events A, B and C are independent events and P(A) = P(B) = P(C) = p, then find the value of P(A ∪ B ∪ C) in terms of p.

Answer»

Events A, B and C are independent events.

P(A) = P(B) = P(C) = p

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P (A ∩ B) – P (A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C)

= P(A) + P(B) + P(C) – P(A) ∙ P(B) – P(A) ∙ P(C) – P(B) ∙ P(C) + P(A) ∙ P(B) ∙ P(C)

= p + p + p – (p × p) – (p × p) – (p × p) + (p × p × p)

= 3p – p² – p² – p² + p³

= 3p – 3p² + p³

= p (3 – 3p + p²)

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If events A, B and C are independent events and P(A) = P(B) = P(C) = p, then find the value of P(A ∪ B ∪ C) in terms of p.

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