Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (b)/Solution 1

There are 3 ways to walk from B to B in two steps:

$$B\rightarrow B\rightarrow B,\qquad B\rightarrow A\rightarrow B,\qquad B\rightarrow C\rightarrow B.$$

Each individual path has probability $p_{2,i}\cdot p_{i,2}$, for $i=1,2,3$, and where $p_{i,j}$ is the probability of going from vertex $j$ to vertex $i$ (which is also the $(i,j)$ entry of the transition matrix computed in the previous question). The total probability of starting at B and ending at B after two steps is the sum

$$p_{2,1}\cdot p_{1,2}+p_{2,2}\cdot p_{2,2}+p_{2,3}\cdot p_{3,2}=0.2\cdot 0.2+0.6\cdot 0.6+0.4\cdot 0.2=0.48,$$

which is the (2,2) entry of the transition matrix squared.