1. Find P(AU(B UC)) in each of the following four cases: (a) A, B, and C...
Find P(A U (Be UC)9) in each of the following four cases: (a) A, B, and C are disjoint events and P(A) 1/2. (b) P(A)2P(BC)3P(ABC)-1/2 (c) P(A)1/2, P(BC) 1/3, and P(AC)0 (d) PA n (BC UC) 0.7
Set operations and probabilities: Find the value of ?(?∪(??∪??)?) for each of the following cases: The events ? , ? , ? are disjoint events and ?(?)=2/5 . ?(?∪(??∪??)?)= ?? The events ? and ? are disjoint, and ?(?)=1/2 and ?(?∩?)=1/4 . ?(?∪(??∪??)?)= ?? ?(??∩(??∪??))=0.7 . ?(?∪(??∪??)?)= ??
2. Prove the following propositions (a) Proposition 1: For every event A, AC A (b) Proposition 2: If A, B, C are events, if A c B and if Bc C, then Ac C (c) Proposition 3: φ-Ω and 0° = φ (d) Proposition 4: If A1, ..Ak (e) Proposition 5: If A and Bare events, then P(A UB)-P(A)+P(B) - P(AB) are disjoint events, then P(UK 1 A.)-Σ'm P(A)
3. Find the product. a) 3p'(2p +5p)(p +2p+1) b) p(3p+7)(3p-7) c) -(4r-2) d) (2a+3)(a -a' +a-a+ 1)
2./ (HINT: DON'T EVEN TRY THIS WITHOUT DRAWING A VENN DI- 0, BC AGRAM.) W, have ftir events with the following informations: P(AU C) = 0.8, /"(A) = 0.5 P(C) = 0.7 and P(B)--02. AB B. - a./ (4 points) Make a Venn diagram. b./ (4 points) Find P(AC). c./ (4 points) Find P(AC) and P(AB) d./ (4 points) Find P(B1A) and P(C1A).
Consider the following probabilities: P(AC) 0.57, PB = 0.36, and P(A n B) 0.03 a. Find P(A | BC). (Do not round intermediate calculations. Round your answer to 2 decimal places.) P(A | BC) b. Find P(BC | A). (Do not round intermediate calculations. Round your answer to 3 decimal places.) P(BC A) c. Are A and B independent events? Yes because PAI B = PA) Yes because PAN B)0 No because P(A I B)PA). No because PAN B)0
Answer the following with TRUE or FALSE, and justify your answer (this does not necessarily have to be a formal prooW). (a) If two events A, B where P(A) > 0, P(B) > 0, P(AN B) > 0 and A, B are independent, then AC and BC are independent as well. (b) For two events A, B where P(A) > 0, P(B) > 0, they can be disjoint and independent at the same time. (c) For two events A1, A2...
(7) The events A and B are mutually exclusive (disjoint). If P(A) = 0.7 and P(B) = 0.2, what is P(A or B)? A) 0.14 B) 0 C) 0.9 D) 0.5 (8) The events A and B are mutually exclusive (disjoint). If P(A) = 0.2 and P(B) = 0.1, what is P(A and B)? A) 0.02 B) 0 C) 0.5 D) 0.3 (9) A probability experiment is conducted in which the sample space of the experiment is S = {1,...
Two events A and B are such that P(A)2, P(B).3, and P(AUB) -4. Find following: a P(An B) b P(AUB) d P(AB) If A and B are independent events with P(A)and P(B) .2, find the following: a P(AUB) b PAnB) c P(AU B)
5. Suppose A, B are events such that P(A) = 1/3, P(B) = 1/4, find P(AUB) under each of the following assumptions: (a) If A and B are mutually exclusive (disjoint). (b) If A and B independent.