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P(A) + P(B) 3. Assume that any conditioning event has positive probability. Prove: P(ANBNC) = P(A|BnC)P(B|C)P(C).

P(A) + P(B) 3. Assume that any conditioning event has positive probability. Prove: P(ANBNC) = P(A|BnC)P(B|C)P(C).

A First Course in Probability (10th Edition)
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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Name: ID Number 1. Two pennies, one with P(head) = u and another one with P(head) = w, are to be tossed together independently. Define Po = P(0 head occurs), P1 P(1 head occurs), Pa = P(2 heads occur). %3D %3D Can u and w be chosen such that po P1 = P2 ? Prove your answer. 2. Assume that any conditioning event has positive probability. Prove that, if A and B are disjoint, then P(A) P(A) + P(B) P(A|AU B) = 3. Assume that any conditioning event has positive probability. Prove: P(ANBNC) = P(A|BNC)P(B| C)P(C). 4. Bowl A contains 60 black chips and 40 white chips. Fifty of these 100 chips are selected at random and without replacement and put in bowl B, which was originally empty. Find the probability that 20 black chips and 30 white chips are transferred from bowl A to bowl B. 5 Find the probability of the union of the events {a < X <b, -x < X2 < x) and (-x< X, < x. c< X < d}. if X and X2 are two independent real-valued random variables with P(a< X, <b) 2/3 and P (e< X2< d) = 5/8. 6. If p(r1.12)- (2/3)****(1/3)- () - (0.0). (0.1).(1.0). (1.1), zero elswhere, the joint PMMF (probability mass function) of X, and N. find the joint PMF of Y-X X, and Y, X + X 7 Let fa - 0 1.olsewhere, and f2(rg)-od01. br denate, reportivh the altioal PDF paolsblits density fnetina) of X given e l pfof A Demm the stants and
Transcribed Image Text:Name: ID Number 1. Two pennies, one with P(head) = u and another one with P(head) = w, are to be tossed together independently. Define Po = P(0 head occurs), P1 P(1 head occurs), Pa = P(2 heads occur). %3D %3D Can u and w be chosen such that po P1 = P2 ? Prove your answer. 2. Assume that any conditioning event has positive probability. Prove that, if A and B are disjoint, then P(A) P(A) + P(B) P(A|AU B) = 3. Assume that any conditioning event has positive probability. Prove: P(ANBNC) = P(A|BNC)P(B| C)P(C). 4. Bowl A contains 60 black chips and 40 white chips. Fifty of these 100 chips are selected at random and without replacement and put in bowl B, which was originally empty. Find the probability that 20 black chips and 30 white chips are transferred from bowl A to bowl B. 5 Find the probability of the union of the events {a < X <b, -x < X2 < x) and (-x< X, < x. c< X < d}. if X and X2 are two independent real-valued random variables with P(a< X, <b) 2/3 and P (e< X2< d) = 5/8. 6. If p(r1.12)- (2/3)****(1/3)- () - (0.0). (0.1).(1.0). (1.1), zero elswhere, the joint PMMF (probability mass function) of X, and N. find the joint PMF of Y-X X, and Y, X + X 7 Let fa - 0 1.olsewhere, and f2(rg)-od01. br denate, reportivh the altioal PDF paolsblits density fnetina) of X given e l pfof A Demm the stants and
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