MathDB

An ant crawls along the grid lines of an infinite quadrille notebook. One grid point is marked red, this is its starting point. Every time the ant reaches a grid point, it continues forward with probability

\frac13

, left with probability

\frac13

, and right with probability

\frac13

. What is the chance that it is after its third turn, but not after its fourth turn that it returns to the red point? If the answer is

\frac{p}{q}

, where

p

and

q

are coprime positive integers, then your answer should be

p + q

. The steps of the ant are independent.

Problem Statement