(2) Given a sample space 2 and a probability measure P, two events A C and...

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(2) Given a sample space 2 and a probability measure P, two events A C and BCN are said to be independent if P(AB) = P(A)P(B). Assume that the events E1, E2 are independent. a) Prove that the events Ei, Es are also independent. b) If, in addition, P(E₁)= and P(E2) = 1, Prove that P(E₁ UE₂) = 17 24 2 = c) Let E3 be a third event such that P(E3) 4, satisfying in addition that E₁ and E3 are independent and also that E2 and E3 are independent. Prove that <P(E₁ UE2 U E3) ≤ 19 24 (2) Given a sample space 2 and a probability measure P, two events A C and BCN are said to be independent if P(AB) = P(A)P(B). Assume that the events E1, E2 are independent. a) Prove that the events Ei, Es are also independent. b) If, in addition, P(E₁)= and P(E2) = 1, Prove that P(E₁ UE₂) = 17 24 2 = c) Let E3 be a third event such that P(E3) 4, satisfying in addition that E₁ and E3 are independent and also that E2 and E3 are independent. Prove that <P(E₁ UE2 U E3) ≤ 19 24