Suppose that events 𝐴, 𝐵, and 𝐶 are independent. Use the definition of independence to show that 𝐴 and 𝐵 ∪ 𝐶 are independent.

Question

. Suppose that events 𝐴, 𝐵, and 𝐶 are independent. Use the definition of independence to show that 𝐴 and 𝐵 ∪ 𝐶 are independent.

Answer 1

A,B,C are independent events implies

P(A∩B)=P(A).P(B)

P(B∩C)=P(B).P(C)

P(C∩A)=P(C).P(A)

P(A∩B∩C)=P(A).P(B).P(C)

now we have top prove P(A∩(B∪C))=P(A).P(B∪C)

solving R.H.S.

P(A).P(B∪C)=P(A)[(P(B)+P(C)-P(B∩C)]

P(A)[P(B)+P(C)-P(B).P(C)]

P(A).P(B)+P(A).P(C)-P(A).P(B).P(C)

P(A∩B)+P(A∩C)-P(A∩B∩C)

P(A∩(B∪C))

hence proved