TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW01.1

Random walk of a robot

A robot is placed at the origin (the point $(0,0)$) on a two-dimension integer grid. Denote the position of the robot by $(x,y)$. The robot can either move right to $(x+1,y)$ or move up to $(x,y+1)$.

Suppose each time the robot randomly moves right or up with equal chance. What is the probability that the robot will ever reach the point $(8,6)$?

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Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

Robot needs to get to $(8,6)$ in any pattern within the first 14 steps.

1) How many paths in 14 steps are possible at all? We can go either right or up:

possible <- 2^14 # possible sequences of up or right
possible

2) How many permutations with 8 steps to the right and 6 steps up exists? How can we choose 8 positions for right steps, the rest is up?

fortunate <- choose(14, 8)
fortunate

3) Now the probability of the robot reaching the point is the number of fortunate divided by the possible cases

(fortunate / possible) * 100

Probability is about 18.3%

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• robot grid.png (details)