Suppose four active nodes-nodes A, B, Cand D-are competing for access toa channel using slotted ALOHA. Assume each node has an infinite number ofpackets to send. Each node attempts to transmit in each slot with probabilityp. The first slot is numbered slot 1, the second slot is numbered slot 2, and soon.a. What is the probability that node A succeeds for the first time in slot 5?b. What is the probability that some node (either A, B, C or D) succeeds in slot4?c. What is the probability that the first success occurs in slot 3?d. What is the efficiency of this four-node system?

Answered: Image /qna-images/answer/b26fb329-59b7-4886-a17d-931b5c0a36c8.jpg

Answered: Suppose four active nodes-nodes A, B, C…

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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Q1) A specific article is produced by machines M, M, M,and M. The probability that a random article is produced by M, Mə and Ma is 0.2,0.3 and 0.1 respectively. The probability for each machine to produce defective articles is given as M, = 0.03, M2 = 0.02, M, = 0.04 and Ma 0.01. What is the probability that a random article was made by Ma, given that it is non-defective?
A corporation has two construction contracts that are to be assigned to one or more of the three bidding firms say A, B, C. (It is possible for one firm to get both contracts). If all outcomes are equally likely, find the probability that both contracts go to the same firm.
An urn contains 10 marbles, of which five are green, two are blue, and three are red. Three marbles are to be drawn from the urn, one at a time without replacement. What is the probability that all three marbles drawn will be green?
A data from a college shows that a certain exam has 10% passing rate. If 4 students take this exam, Find: a) probability that all of them pass it b) probability that no one pass it c) probability that at least one of them fails
At the entrance to an online game, there are two games. The first game is programmed so that in the long run it will produce a winner in 10% of the plays. The second game is programmed so that in the long run it will produce a winner in 15% of the plays. a.) If we play each game once, what is the probability that we will win on both plays? b.) If we play each game once, what is the probability that we will lose on both plays? c.) If we play each game once, what is the probability that we will win on at least one play?
2. Use Bayes' theorem to solve the following: There are two bags. The first bag contains four mangoes and two apples; the second bag contains four mangoes and four apples. We also have a biased coin, which shows "Heads" with probability 0.6 and "Tails" with probability 0.4. If the coin shows "Heads". we pick a fruit at random from bag 1; otherwise we pick a fruit at random from bag 2. Your friend flips the coin (you cannot see the result), picks a fruit at random from the corresponding bag, and presents you a mango. What is the probability that the mango was picked from bag 2?
2. A group of three undergraduate and six graduate students are available to fill certain student government posts. If four students are randomly chosen from this group, find the probability that exactly two undergraduates will be among the four chosen. [Hint: How many ways are there to fill the posts? How many ways satisfy the condition of having exactly TWO undergraduates selected AND hav- ing exactly two graduates selected?]
A k out of n system is one in which there is a group of n components, and the system will function if at least k of the components function. Assume the components function independently of one another. a) In a 3 out of 5 system, each component has probability 0.9 of functioning. What is the probability that the system will function? b) In a 3 out of n system, in which each component has probability 0.9 of functioning, what is the smallest value of n needed so that the probability that the system functions is at least 0.90?
In the following diagram, each switch is in a closed state with probability p, and in the open state with probability 1-p. Assuming that the state of each switch is independent to each other. Find the probability that a closed path can be maintained between A and B. ] B
A circuit has three independent switches A, B , C wired inparallel. current can only flow through the bank of switches if at least one of them is closed. The probability that any given switch is closed is 0.9. calculate the probability that the current can flow throught the bank of switches.

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