 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii/5/b/i/solution

There are two paths from

X

back to

X

in two steps:

X \to X \to X

and

X \to Y \to X

. By the markov property, their probabilities multiply along each path. Therefore

p_{2} = \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{3}{8} = \frac{1}{4} + \frac{3}{16} = \frac{7}{16} .

 New to topics? Read the docs here!