problem 54

(a)

p_{k} = p p_{k - 1} + q p_{k + 1}

with boundary condition

p_{0} = 1 .

(b)

Let

A_{j}

be the event that the drunk reaches

k

before reaching

- j

. Then,

A_{j} \subseteq A_{j + 1}

since to reach

- (j + 1)

the drunk needs to pass

- j

. Note that

\infty ⋃ j = 1 A_{j}

is equivalent to the event that the drunk ever reaches

k

, since the complement of this event, namely the event that the drunk reaches

- j

before reaching

k

for all

j

implies that the drunk never has the time to reach

k

By assumption, $P (\infty ⋃ j = 1 A_{j}) = lim n \to + \infty P (A_{n}) .$ $P (A_{n})$ can be found as a result of a gambler’s ruin problem.

If $p = \frac{1}{2}$ ,

P (A_{n}) = \frac{n}{n + k} \to 1 .

If $p > \frac{1}{2}$ ,

P (A_{n}) = \frac{1 - {(\frac{q}{p})}^{n}}{1 - {(\frac{q}{p})}^{n + k}} \to 1 .

If $p < \frac{1}{2}$ ,

P (A_{n}) = \frac{1 - {(\frac{q}{p})}^{n}}{1 - {(\frac{q}{p})}^{n + k}} \to {(\frac{p}{q})}^{k} .