Past exam of the options I course of the University of Oxford / 2011 / M / 2 / a

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- Solution a

Solution

Condition on the intermediate state. Starting at

+ 1

, a return to

+ 1

after two transitions occurs through

- 1

with probability

q (1 - p)

or through

+ 1

with probability

(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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