2. Suppose A, B, and C are events of strictly positive probability in some probability space. If ...
Stochastic Processes 4 Consider a probability space (2, *, P) and assume that the various sets mentioned below are all in (a) Show that if D, are disjoint and P(C | D)-p independently of i, then P(C I U,D -p. (c) Show that if E, are disjoint and U, E -, then (d) Show that if C, are disjoint and PAC) P(BIC) for all i,
The events A, B and C form a partition of the sample space 2. Suppose that we know that P(A U B) 5/8 and that P(B U C) 7/8. Find P(A) P(B) and P(C); explain how you arrive at your answers.
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)
Suppose A and B are events in a sample space Ω. Let P(A) = 0.4, P(B) = 0.5 and P(A∩B) = 0.3. Express each of the following events in set notation and find the probability of each event: a) A or B occurs b) A occurs but B does not occur c) At most one of these events occurs
Please anyone can help me with this probability question and please provide explanation. Thank you so much! (i) Consider two events A and B, with P(A) = 0.3, P(B) = 1. Compute P(A∩B), P(Ac ∩B), and P(A ∩ Bc) (where we denote by Ac = Ω \ A the complement of an event A). (ii) We now consider three independent events A, B, and C. Using the definition of inde- pendence, show that the two events Ac and Bc ∪...
Suppose A and B are independent events. In expression (1.4.6) we showed that Ac and B are independent events. Show similarly that the following pairs of events are also independent: (a) A and Bc and (b) Ac and Bc
4. (a) Show that (b) Two events A and B are said to be conditionally independent given C if P(An BIC)P(A|C)P(BC). Prove that if A and B are conditionally independent given C, then
Events A and B are defined on the same sample space that has 20 outcomes. A∩B includes 3 outcomes, AC∩B includes 5 outcomes and A∩BC includes 6 outcomes. How many outcomes are in (A∪B)C?
Assume that A and B are events in a probability space with the property that P(A) = 0.5, P(B) = 0.6, and P(A ∪ B) = 0.9. 1. Explain why A and B cannot be independent. 2. Is A favorable or unfavorable to B? (Remember that an event E is said to be favorable to F if P(F|E) > P(F); that is, if the knowledge that E occurred increases the plausibility of F.)
Problem 3. Let A, B, C and D be four events, each with positive probability. 1. Prove the 'chain rule of conditional probability: P(An BnCAD) - P(A) P(BA) P(CAN B) P(DjAnBnC). 2. Prove an extended version of Bayes formula: P(ABC)= LARO _P(BLA) P(CBA) P(A) P(B) P(C|B) 1 3. Let E1, E2, E3 form a partition of S2. Prove that P(B) = P(Q(EU E)) + P(BNE).