 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2011/m/2/a/solution

Condition on the intermediate state. Starting at

+ 1

, a return to

+ 1

after two transitions occurs through

- 1

q (1 - p)

or through

+ 1

(1 - q)^{2}

. These alternatives are disjoint, so

P (X_{3} = + 1 ∣ X_{1} = + 1) = q (1 - p) + (1 - q)^{2} .

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