 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/12/a/solution

Use independent choices with constant probabilities

p

and

q = 1 - p

. Completing the entire reverse circuit while the forward-moving object never moves requires the first

F

choices all to be reverse moves. Thus

P = q^{F} = (1 - p)^{F} .

Here and below, a zero-length product is interpreted as one, including at

p = 0

p = 1

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