Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 7H / b / ii

Table of contents
- Solution ii

Solution

Let

p_{n} = P (X_{n} = A)

. Whenever the taxi is away from the airport it returns with probability

3/4

, whereas it must leave whenever it is at the airport. Hence

p_{n + 1} = \frac{3}{4} (1 - p_{n}), p_{0} = 1.

Subtracting the fixed point

3/7

gives

p_{n + 1} - \frac{3}{7} = - \frac{3}{4} (p_{n} - \frac{3}{7}) .

Therefore

p_{n} = \frac{3}{7} + \frac{4}{7} (- \frac{3}{4})^{n}

for every

n \geq 0

, and in particular for the requested

n \geq 1

.

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