3. Let A, B, C be events in a sample space S. Prove that (a) P(AUB)...
Exercise 4. By writing AU BUC as (AUB) UC, show that the Principle of Inclusion-Exclusion for three sets is P(AUBUC) = P(A)+P(B)+P(C)- P(ANB) - P(ANC) - P(BNC)+P(ANBNC) Can you generalize the result to an arbitrary number of events?
1. Let A, B and C be events in the sample space S. Use Venn Diagrams to shade the areas representing the following events (32 points) a. AU (ANB) b. (ANB) U ( AB) C. AU ( ANB) d. (AUB) N (AUC)
1. (15pts) Events A, B and C are such that P(A) = 0.7, P(B) = 0.6, P(C) = 0.5, P(AnB) = 0.4 , P(AnC) = 0.3, P(BnC) = 0.2, P(AnBnC) Find (a) either B or C happens (b) at least one of A, B, C happens; c) exactly one of A, B, or C happens. 0.1.
Let and B be events in a sample space S, and let C = S - (AUB). Suppose P(A) = 0.8, P(B) = 0.2, and P(An B) = 0.1. Find each of the following. (a) P(AUB) (b) P(C) (c) PAS (d) PLAC BC) (e) PLACUBS (1) P(BCnc)
6. (10 points) Let A, B, and C be sets. Prove (AuB)C(AnC) u(BnC)
05 (24 marks) Let A, B, and C be three events in the sample space S. Suppose we know that A U B U C-S, P(A)-1/2, P(B)-1/3, PALJ B-3/4. Answer the following questions: a) Find P(AnB). (4 marks) b) Do A, B, and C form a partition of S? Why? (4 marks) c) Find P(C-(AUB)). (8 marks) d) If P(Cn (AU B))-5/12, find P(C). (8 marks)
Please help me prove 2,4, and 5. Thank you Theorem 17. Let A, B and C be sets. Then the following statements are true: (1) AB CA; (2) B CAUB; (3) A CAUB; (4) AB=BA; (5) AU (AUC) = (AUB) UC; (6) An(BNC) = (ANB) nC; (7) An (BUC) = (ANB) U (ANC); (8) AU (BAC) = (AUB) n(AUC).
Problem 1. Let A event from outcome space S,equipped with probability function I . Prove that P(A) 1. Hint: You can use theorem 1.4 Problem 2. Let A,B,C events from outcome space S, equipped with probability function P. Prove that P(AUBUC)- P(A)+P(B)+P(C)- PAnB) PAnC) PBnC) +P(AnBnc) Hint: You can treat A and BUC as two events and apply theorem 1.6. You will also need to use Law 5 from the distributive laws.
Problem 1. Let A event from outcome space S,equipped with probability function I . Prove that P(A) 1. Hint: You can use theorem 1.4 Problem 2. Let A,B,C events from outcome space S, equipped with probability function P. Prove that P(AUBUC)- P(A)+P(B)+P(C)- PAnB) PAnC) PBnC) +P(AnBnc) Hint: You can treat A and BUC as two events and apply theorem 1.6. You will also need to use Law 5 from the distributive laws.
In a sample space, events A and B are independent, events B and C are mutually exclusive, and A and C are independent. a) Show that P(AUB) = P(B) + P(A)P(B') = P(A) + P(A')P(B) b) If P(AUBUC) = 0.9, P(B) = 0.5 and P(C) = 0.3 find P(A).