An urn initially contains 2 balls, one of which is red and the other blue. At each

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An urn initially contains 2 balls, one of which is red and the other blue. At each stage a ball is randomly selected. If the selected ball is red, then it is replaced with a red ball with probability 0.7 or with a blue ball with probability 0.3; if the selected ball is blue, then it is equally likely to be replaced by either a red or blue ball.

(a) Let Xn equal 1 if the nth ball selected is red, and let it equal 0 otherwise.

Is {Xn,n ≥ 1} a Markov chain? If so, give its transition probability matrix.

(b) Let Yn denote the number of red balls in the urn immediately before the nth ball is selected. Is {Yn,n ≥ 1} a Markov chain? If so, give its transition probability matrix.

(c) Find the probability that the second ball selected is red.

(d) Find the probability that the fourth ball selected is red.

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Hitarth Singh

I did my Bachelor's in Computer Science from Delhi University. I am currently pursuing my Master's in Computer Science from Chennai Mathematical Institute, Chennai, India. My current research interest lies in program verification and synthesis. During my undergraduate studies, I used to work as a home tutor for Maths and Computer Science. I have been working as a freelance tutor for the last two years. My teaching experience has been incredible, and I think it helped me learn as much as it helped my students. I also try to help my classmates and juniors with studies by giving them formal/informal tutorials, and this also helps me to become an effective and better teacher. Almost all of my students admired my teaching and gave positive feedback. I try to teach from the level the student is most comfortable with. The students usually struggle with their projects, assignments and subject material, and I always try my best to help and motivate them, so they can confidently and correctly complete their work. I understand the academic and work pressure the students are in, and I have seen that with proper guidance they can easily, and happily, excel in their courses. While teaching, if possible, I usually focus on problem-solving as it is the best way to clearly understand difficult concepts. When answering a question, I make sure to give them plenty of examples, and also give the necessary background and explain fundamentals related to that concept. Whenever relevant, I try to provide the learner with the history and interesting trivia related to the concept to make the teaching more interesting, and hence more effective. I also teach web development. I have worked as a freelance web developer, and I've successfully hosted many commercial websites. While teaching web development, my strategy is to give hands-on examples. I usually teach web development by setting up a goal for the learner of creating a complete web application. With that in hand, I split the task into modules, and I start implementing each module with the student and let them stop me, and start developing themselves when they feel enough confidence that they can finish it.

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