5. Show that the conditional independence of A and B given C neither implies, nor is implied by,the independence of A and B. For which events C is it the case that, for all A and B, the events A andB are independent if and only if they are conditionally independent given C?

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Answered: 5. Show that the conditional…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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7. If A, B, C are independent events, show directly that both A UB and A\B are independent of C.
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9. Assume that A and B are independent events. Show that the following pairs of events are also independent: (a) A and Bº; (b) Ac and B; (c) Ac and Bc.
Let us have events/sets A and B such that P(A)=0.4 and P(B)=0.3. Is it possible for A and B be mutually exclusive/disjoint events?
Choose the following statements that are true for any events A, B. O Pr{AU B} = Pr{A}+ Pr{B} Pr{ANB} O Pr{AU B} = Pr{A} + Pr{B} O Pr{AU B} Pr{A} + Pr{B} O Pr{A} < Pr{ANB} O Pr{A} < Pr{AUB}
{1,2,3,4,5,6} 2. Let P(A)=0.3, P(B) = 0.9 and P(ANB) = 0.36, then the events A, B are A. Independent C. Mutually Exclusive 3. The B. Not Independent D. None
1. I. The events A and B are mutually exclusive. If P(A) = 0.1 and P(B) = 0.3 , what is P(A or B)? II. The events A and B are mutually exclusive. If P(A) = 0.1 and P(B) = 0.5, what is P(A and B)? III. suppose that A and B are two events. If P(A and B) = 0.28 and P(A) =0.9, what is P(B|A)
B1. Consider events A, B and C, where A and B are mutually exclusive. Using multiple Venn diagrams, determine if the following proposition is correct, or not: (ANB)U(ANC) = (An BenC)u(ANBNC).
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If two events are mutually exclusive, then: A They cannot be complements. B) the events can occur at the same time They also could be complements. they are independent
Example 3-28. Let P (A) = p, P (A \B) = q, P (B | A) = r. Find relations betwe number p, q, r for the following cases : (a) Events A and B are mutually exclusive. (b) A and B are mutually exclusive and collectively exhaustive. (c) A is sub-event of B ; B is a sub-event of A. (d) Ā andB are mutually exclusive.
let A and B be two events in an outcome space M such that M = A U B, P (A) = 0.65 and P (B) = 0.75.Calculate P (A ∩ B)Are events A AND B independent?

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5. Show that the conditional independence of A and B given C neither implies, nor is implied by, the independence of A and B. For which events C is it the case that, for all A and B, the events A and B are independent if and only if they are conditionally independent given C?