Please help me with this probability and statistics question A robot is located
ID: 3306773 • Letter: P
Question
Please help me with this probability and statistics question
A robot is located at position (0,0) on a grid, and want to move to position (20, 10). However, it can only move along the grid lines towards the top or right, i.e., being at position (i,j), the next point to be is either (i.j +1) (moving towards the top) of (i1j) (moving towards the right). Two possible paths are shown in the figure below. (20,10) (0,0) (a) How many different paths exists for the robot to reach the destination? (b) Assume the robot picks one of the possible paths randomly at random (equal probability). What is the proba- bility that the robot passes the point (10,5) along its path towards the destination? (c) Now, consider the following random experiment. In the first step, the robot toss a fair (PHPIT] 1/2) coin, and moves to the right if the outcome is a head, and moves to the top otherwise. Then for the rest of the path, it chooses one of the possible paths to go to the destination with uniform probability. Under this random experiment, what is the chance that the robot passes through the point (10,5)?Explanation / Answer
a) total number of right steps =20 and top steps =10 ; hence total number of steps =20+10 =30
total number of paths =total number of ways to distribute 30 into two parts of 20 and 10 =30!/(20!*10!) =30045015
b)similarly number of ways to move from (0,0) to (10,5) = (15!/(10!*5!)) =3003
number of ways to move from (10,5) to (20,10) =(15!/(10!*5!)) =3003
therefore possible paths that passes through (10,5) =3003*3003
hence probability that it will passes through point (10,5) =3003*3003/30045015 =0.30015
c) for above probability will remain same as before hand also it has equal probability to move to top or right.
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