Assume that a website www.funwithmath1600.ag has three pages: • Page A: KingAlgebra • Page B: Learn1600andWin • Page C: LinearAlgbraIsEverywhere Each page has some links to the other pages of this website and no pages links to any page outside this website. • Page A has three links to page B and only one link to page C. • Page B has three links to page A and two links to page C. • Page C has one link to page A and two links to page B. A student decides to explore this website starting from page A. Since reading content is always a boring task (is it?!) they decide to choose one of the links in page A with equal probability and click on the link to see the next page. As a result, on the next step, they will end up on page B with probability 3/4 and on the page C with probability 1/4 . This process is then continued by the student with the same rule: Go the next page by clicking, with equal probability, on one of the existing links that are on the present page. (Use only fractions in your calculations; no decimals please). (a) Use a Markov chain to model the probability of the student being on one of the pages of this website after n clicks. In particular, find the transition matrix, the initial state vector and explain the meaning of entries on each state vector. (b) Use your model to find the probability that the student ends up on page B after 5 clicks. (c) Find the steady state of this stochastic process. (d) Explain why one can use this information to rank the pages of this website and what would the ranking be?
Assume that a website www.funwithmath1600.ag has three pages:
• Page A: KingAlgebra
• Page B: Learn1600andWin
• Page C: LinearAlgbraIsEverywhere
Each page has some links to the other pages of this website and no pages links to any page outside this website.
• Page A has three links to page B and only one link to page C.
• Page B has three links to page A and two links to page C.
• Page C has one link to page A and two links to page B.
A student decides to explore this website starting from page A. Since reading content is always a boring task (is it?!) they decide to choose one of the links in page A with equal probability and click on the link to see the next page. As a result, on the next step, they will end up on page B with probability 3/4 and on the page C with probability 1/4 . This process is then continued by the student with the same rule:
Go the next page by clicking, with equal probability, on one of the existing links that are on the present page.
(Use only fractions in your calculations; no decimals please).
(a) Use a Markov chain to model the probability of the student being on one of the pages of this website after n clicks. In particular, find the transition matrix, the initial state vector and explain the meaning of entries on each state vector.
(b) Use your model to find the probability that the student ends up on page B after 5 clicks.
(c) Find the steady state of this stochastic process.
(d) Explain why one can use this information to rank the pages of this website and what would the ranking be?
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