Question

A robot is standing at the origin (0,0) of a square grid. The robot is programmed to move exactly one unit at a time, in one of three directions: up, down, and right. The robot will never move left. For example, starting at (0,0), the robot will move either • one unit up, to (0,1), one unit down, to (0,-1), or • one unit right, to (1,0). . The robot is programmed to make exactly 5 moves, and then stop. (a) How many different 5 move paths can the robot make? Note: Paths are considered different if they differ at all. For example, . Up, Up, Right, Up, Down Up, Up, Right, Down, Up . are considered to be two different paths, even though they overlap for the first 3 moves, and the robot ends up in the same place at the end. (b) How many different 5 move paths can the robot take to end up at (3,0)? (c) For each move, the robot is programmed to pick a direction uniformly at random. So for each move, independently of all other moves, the robot has a 1/3 probability to move up, a 1/3 probability to move down, and a 1/3 probability to move right. What is the probability the robot will end up at the point (3,0) after completing its 5 moves?

Answer 1

© Right, Right, up, Right, down 1(1,0), (2,0), (2,), (3,1), 13,0)) ot, Squwie grid Unit one → We 5 4 3 2 © Up, Right, down, Right, Right. ((0,1), (1,1),(1,0), (2,0), (3,0)) Right, Right, Right, up, down ((0,0), (2.0),(3.0),(3,0), (3,0)) © Right, up, Right, right, down (11,0), (1,1),(2,1), (3,1), (3,0)) Solution :-> given 910bot, is standing at the origin of Squwie hobot at a time, and one of three direction: up, down and right → Robot will never move left. @ Now we find different 5 move polhs can the robot make Buch that globot ends up in the same place at the end. make following graph (1,9) 0100,0) stårl with 10,0) Ale possible 5 move paths lathafn giobot make and end at (1,2). UP, UP, Right, Up, Down (10,1),(0,2), (1,2), (1,3), (1,2) Ý UP, UP, Right Down, up ((0,1),(0,2), (1,2), (1, 1), (1,2)) © UP, UP, UP, Right, Down (10,1),(0,2),(0,3), (1,3), (1,2)) UP, Right, Up, op, Down ((0,1),(1,1),(1,2), (1,3),(1,2)) S UP Pight, docon, up, up ((0,1), (1,1),(1,0), (1,1),(1,2)) © Right, UP, UP, UP, down ((1,0), (1,1), (1, 2), (1,3), (1,2)) Now, we find different s move paths can the robot take to end up at (3,0) which is start with (0,0) -> from above graph total possible paths are below, 1 + (3,0) 1 2 5 1

nove UP © giver fach moves are prete pendent with probability of is to the Right move Ź , down move ſ. > NOW, probability, the robot will end up at the point (3,0) after completeing its 5 moves multiplication of probability of each 5 moves 5 times multiplication of = tx tx / XLX - probability of independent events is equals to multiplication of individual eo probability the robot will end up at the point (3,0) after 5 moves 3 3 1 243. events probability 243

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