A roomba is small, disc-shaped. (ideally) autonomous vacuum cleaner that navigates around the floor of a house to clean it. You purchased a heavily discounted model online. After a few weeks, you...

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Extracted text: A roomba is small, disc-shaped. (ideally) autonomous vacuum cleaner that navigates around the floor of a house to clean it. You purchased a heavily discounted model online. After a few weeks, you suspect that it's only moving around randomly rather than using sensors to detect dirty spots on the floor. You gather data on its movements in a room with patches of dust scattered around. Analyzing your results. you conclude that it operates by moving either forward, left, or right (relative to the current direction it is moving) for exactly two seconds before changing direction again, regardless of the presence or absence of dirty spots. Furthermore, the probability of it moving in one of these directions is affected solely by what relative direction it last chose. If previously moving forward, its next move has an equal probability to be any of the 3 directions. If previously moving either left or right, its next move has a 50% chance to be forward and 25% chance each to be either left or right. • What is the transition matrix P? • Out of all decisions it makes in an open space, what percentage of the time is the roomba expected to move forward?