Suppose that 100 students take a test in a class. Each student has a 2/3 probability of passing the test, and the passing of the test is independent over the students. You wish to bound (from above) the probability that no more than 30 of the students pass the test. a. Use the Markov inequality to bound this probability. b. Use the Chebyshev inequality to bound this probability. c. Use the Chernoff bound to bound this probability.

Suppose that 100 students take a test in a class. Each student has a 2/3 probability of passing the test, and the passing of the test is independent over the students. You wish to bound (from above) the probability that no more than 30 of the students pass the test. a. Use the Markov inequality to bound this probability. b. Use the Chebyshev inequality to bound this probability. c. Use the Chernoff bound to bound this probability.

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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Question 1.16.2 (Markov chains, Part IB, 1991, 307D) Three girls A, B and C are playing table tennis. In each game, two of the girls play against each other and the third girl does not play. The winner of any given game n plays again in game n + 1. The probability that girl x will beat girl y in any game that they play against each other is sx/(Sx+Sy) for x, y = {A,B,C}, x‡y, where sa, SB, SC represent the playing strengths of the three girls. In what proportion of games does each girl play, in the long run?
Central topic markov chains: Every summer the Yates de los Lagos owners association decides if its annual regatta will be held in June, July or August. If it takes place in June, the probability of good weather is ¾; and if these conditions exist, the regatta of the next year it will be done in June with probability 2/3, in July with probability 1/6 or in August with probability 1/6; but if there is bad weather, next year's regatta will take place in July or August with equal probabilities. If the competition is held in July, good and bad weather have equal probabilities; if there is good weather, the next year's regatta will be held in July; If there is bad weather, the next regatta will take place in August with probability of 2/3 or in June with probability of 1/3. If the regatta takes place in August, the probability of good weather is 2/5; and if there are good conditions atmospheric conditions, next year's regatta will take place in July or August, with equal probabilities; but…
DO NOT COPY OTHER SOLUTIONS. USE A TRANSITION PROBABILITY MATRIX WHEN SOLVING. Consider the following Markov chain. Assuming that the process starts in state 4, what is the probability that we eventually reach the recurrent class {1,2}?
2. This year Karen has decided to diversify her business by growing broc- coli as well as lettuce. Karen plants lettuce with a probability of or broccoli with a probability of . By the time these plants are ready to harvest the price of lettuce increases with a probability of and decreases with a probability of 3. The price of broccoli increases with a probability of , decreases with a probability of or stays the same with a probability of . (a) Draw a probability tree for this scenario. (b) Use the probability tree to determine the probability that Karen plants broccoli and that the price of broccoli increases by the time it is ready to harvest. (c) Use the probability tree to determine the probability that the price of whatever Karen plants increases by the time it is ready to harvest. (d) Calculate Pr(Price of lettuce does not change | Karen plants lettuce). (e) Calculate Pr(Price of broccoli does not changen Karen plants broccoli).
Show full answers and steps to part d) and e) using Markov Chain Theory. Please explain how you get to the answers without using excel, R or stata
Assume the chances of failure of each component is given in Figure. What is the probability that the system would not work? .
Which statements are true? Select one or more: a. Markov’s inequality is only useful if I am interested in that X is larger than its expectation. b. Chebyshev’s inequality gives better bounds than Markov’s inequality. c. Markov’s inequality is easier to use. d. One can prove Chebyshev’s inequality using Markov’s inequality with (X−E(X))2.
Please do the following questions with handwritten working out. Answer is in other image
ne bets $1. 0.45, he wins the game and with probability 1 - p = 0.55, he loses the game. His goal is to increase his capital to $3, and as soon as he does, the game is over. The game is also over if his capital is reduced to zero. Construct an absorbing Markov chain and answer the following questions. • What is the expected duration of the game? • What is the probability that he goes broke?
Consider purchasing a system of audio components consisting of a receiver, a pair of speakers, and a CD player. Let A₁ be the event that the receiver functions properly throughout the warranty period, A₂ be the event that the speakers function properly throughout the warranty period, and A3 be the event that the CD player functions properly throughout the warranty period. Suppose that these events are (mutually) independent with P(A₁) = 0.92, P(A₂) = 0.96, and P(A3) = 0.90. (Round your answers to four decimal places.) (a) What is the probability that all three components function properly throughout the warranty period? (b) What is the probability that at least one component needs service during the warranty period? (c) What is the probability that all three components need service during the warranty period? (d) What is the probability that only the receiver needs service during the warranty period? (e) What is the probability that exactly one of the three components needs service…