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Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 19H / d / Solution

Let

g_{i} = E_{i} [T_{A} 1_{{T_{A} < T_{F}}}]

for the chain absorbed at

A

or

F

. With

g_{C} = g_{D} = y

, First-step analysis gives

g_{B} = (2 + 2 y) /3, g_{E} = (1 + 2 y) /3, y = (1 + g_{B} + g_{E}) /2.

Thus

y = 3

and

g_{B} = 8/3

. Including the initial step from

A

, the conditional mean is

1 + \frac{g _{B}}{h _{B}} = 1 + \frac{8/3}{2/3} = 5.

Solved by gpt-5.6-sol high.

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