ISTANBUL KULTUR UNIVERSITY DEPARTMENT OF INDUSTRIAL ENGINEERING IE 5202 OPERATIONS RESEARCH II SUMMER 2020 - July 21", 2020 MIDTERM - PART IV Duration: 25 minutes Grade: 30 points Jonas Kahnwald is a teenager living in a town called Winden in Germany. He discovered a cave in which there exists a passage that serves as a time machine: This passage allows the passengers to travel between the years 1920, 1953, 1986, 2019, and 2052. Jonas also discovered that the passage works according to the following rules: e . In a single voyage (single travel), a passenger can only travel to the nearest possible year, or he can stay in the year that he is already in. . For the years 1953, 1986, and 2019, the probability that he travels to the nearest possible year is half of the probability he stays in the year that he is in. For example, if the passenger is in 1953: the probability that he will stay in 1953 is 0.5, the probability that he will travel to 1920 is 0.25, and the probability that he will travel to 1986 is 0.25. .If the passenger is in year 1920 or 2052, the probability that he stays in the year that he is in or the probability that he travels to the nearest possible year are equal to each other. For example, if the passenger is in 2052; the probability that he will stay in 2052 is 0.5, and the probability that he will travel to 2019 is also 0.5. Based on Jonas' discoveries: a) Is the above stochastic process discrete or continuous? Explain why. b) Define the states for the above stochastic process, c) Construct one-step probability transition matrix. d) Suppose that Jonas is in 2019 now. What is the probability that he will be in 2019 after two voyages (after two travels)