Problem2: A) Let A and B be two events such that: P(B) = 0.3, P(AUB) =...

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Problem2: A) Let A and B be two events such that: P(B) = 0.3, P(AUB) = 0.7. Answer to the following questions: 1) Compute P(A – B). 2) If P(An B) = 0.1, what is P(A) ? 3) Compute P[(An B)c]. 4) Compute P(Aºn Bc). B) Let A, B, C be some events. Show the following identities. A mathematical derivation is required, but you can use diagrams to guide your thinking. 1) P(AUBUC) = P(A) + P(B) + P(C) - P(An B) - P(BnC) - P(ANC) +P(An BnC). 2) P(AUBUC) = P(B) + P(An Bc) + P(CnAºn Bc) C) Let A and B be two events. Use the axioms of probability to prove the following: 1) P(An B) ≥ P(A) + P(B) – 1. 2) P(An BnC) ≥ P(A) + P(B) + P(C) — 2. 3) The probability that one and only one of the events A or B occurs is P(A) + P(B) – 2P(An B). Problem2: A) Let A and B be two events such that: P(B) = 0.3, P(AUB) = 0.7. Answer to the following questions: 1) Compute P(A – B). 2) If P(An B) = 0.1, what is P(A) ? 3) Compute P[(An B)c]. 4) Compute P(Aºn Bc). B) Let A, B, C be some events. Show the following identities. A mathematical derivation is required, but you can use diagrams to guide your thinking. 1) P(AUBUC) = P(A) + P(B) + P(C) - P(An B) - P(BnC) - P(ANC) +P(An BnC). 2) P(AUBUC) = P(B) + P(An Bc) + P(CnAºn Bc) C) Let A and B be two events. Use the axioms of probability to prove the following: 1) P(An B) ≥ P(A) + P(B) – 1. 2) P(An BnC) ≥ P(A) + P(B) + P(C) — 2. 3) The probability that one and only one of the events A or B occurs is P(A) + P(B) – 2P(An B).

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