(Solved): Problem 14 (Amoeba random walking) In this problem you will work out a simple model for an amoeba's ...

Problem 14 (Amoeba random walking) In this problem you will work out a simple model for an amoeba's quest for dinner. This amoeba lives in a two-dimensional world, on a surface conveniently divided into grids. For our purposes, the surface is infinite in extent, and everywhere the same. Part of the infinite plane is shown at right. In a time interval \( \Delta t \) the amoeba can move exactly one grid to the right (R), one grid to the left (L), one grid up (U), one grid down (D), or it might stay (S) where it is. You may assume the amoeba's decisions are completely random, so that each possibility is equally likely. We begin the observing the amoeba at time \( t=0 \), with the amoeba at point \( X \). A. After a time of only one \( \Delta t \), how many different ways could the amoeba have ended up at each the following points (i.e. how many distinct paths with a length of 1 step end at the given point)? 1. \( \mathrm{A} \) ii. \( \mathrm{B} \) iii. \( \mathrm{C} \) iv. \( \mathrm{x} \) v. How many total paths are associated with one unit of time \( \Delta t \) ? B. After two units of time, (so \( =2 \Delta t) \), how many different ways could the amoeba have ended up at each of the following points? i. \( \mathbf{A} \) ii. \( \mathrm{B} \) iii. \( \mathrm{C} \) iv. \( x \) v. How many total paths are associated with time \( t=2 \Delta t \) ? vi. What is the probability \( P \) that the amoeba will be found at the point \( C \) at time \( t=2 \Delta t \) ?