Problem 4Let C1, C2, ·.., Cm be a partition of the sample space S, and A and B be two events. Suppose weknow that• A and B are conditionally independent given C;, for all i E {1, 2, · . · , M};• Bis independent of all C;'s.Prove that A and B are independent.

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Answered: Problem 4 Let C1, C2,., CM be a…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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26. If P(A) = 0.3 P(B) = 0.2 and P(C)=0.1 and A, B, C are independent events the probability of occurrence of at least one of the three events A,B, C is
Be A, BY C, three events associated with the same experiment. Suppose they meet the following conditions: P(4) =D 글, P(B) = , P(C) 3D , P(AN B) = }, P(AnC) = y P(B n C) = 0 P(An(BUC). 1 6 y P(BnC) = 0 Calculate P (A (E
8. Let A, B be two events with P(A) = ½, P(A − B) = 3, determine P(AB).
4. A spinner is now created with four equal sized sectors as shown, An experiment is run where the spinner is spun twice and the outcome is recorded each time. (a) Create a sample space list of ordered pairs that represent the outcomes, such as (4, 2), which represent spinning a 4 on the first spin and a 2 on the second spin. 3 2 (b) Using your answer from (a), determine the probability of obtaining two numbers with a sum of 4.
5. A candy dish contains 2 red (R) and 4 yellow (Y) candies. You close your eyes, select two candies from the dish (without replacement), and record their colors. Use general multiplication rule and special addition rule to find the following probabilities. a. The probability that the first selected candy will be red and the second selected candy will be yellow is b. The probability that two selected candies will be of different colors_ One is red, another one is yellow c. Probability that two selected candies will be of the same color_ Both are red or both are yellow Assuming that the event "The first selected candy will be red and the second selected candy will be yellow "is denoted by A, the event "Two selected candies will be of different colors" is denoted by B, and the event "Two selected candies will be of the same color" is denoted by C, show your work in finding P(A), P(B), and P(C) in the questions a, b, and c.
8.) Six letters of the alphabet are chosen at random. Find the probability that the following "words" are formed: (a) REDSOX (b) A "word" that ends in Y (c) A "word" with no repeated letters (d) A "word" that does not contain A,B, or C
13. It is given that 5 P (A¬A2) =, P (A2) = - 1 P (A, + A2) = ?, 6' Show that the events Aj and A2 are independent.
4
There are two urns. The first urn has 2 black balls and 1 white ball inside and there are 2 black balls and 2 white balls in the second Urn. You ask your friend to randomly choose an Urn and pick 2 balls one by one with no replacement. 1c) If one black and one white ball are selected, what is the probability that your friend selects the Urn with 3 balls inside? 1d) If we repeat this experiment from the beginning with 5 different friends. What is the probability that 2 of them get 2 black balls? 1e) On average (expected value) how many of those 5 friends will get 2 black balls?
6. Let A, B, C be three events with P(A) = P(B) = P(C) = 1, P(AC) = 1, P(AB) = P(BC) = 0, determine the probability of the event that at least one of A, B, C occurs.
A personnel director has two lists of applicants for a job, List 1contains the name of 5 women and 2 men,whereas list 2 contains thename of 2 women and 6 men. A name is randomly selected fromlist 1 and added to list 2. A name is then randomly selectedfrom the augmented list 2. Given that the name selected fromlist 2 is that of a man what is the probability that a woman's namewas originally selected from list 1.?

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Problem 4 Let C1, C2,., CM be a partition of the sample space S, and A and B be two events. Suppose we know that • A and B are conditionally independent given C;, for all i e {1,2, -, M}; • Bis independent of all C;'s. Prove that A and B are independent.