Consider we have a sequence of random walks such that we have the transition probability matrix as give, which is , but with the difference that we now have two states denoted as -1 and +1 only, such that  and . Given this we are as usual interested to find  and .

Hence:

If we continue doing it we get

 and

Given this we find

(i) ,
in general the formulae would be ,
depending on the number of states, such that there are even number
of positives and equal number of negatives, i.e.,
and we will have the , such that

(ii)
 and in general the formulae would be ,
depending on the number of states such that there are even number of positives and
equal number of negatives, i.e.,  and we will have the
,such that

(iii)

Now we already know that

 and hence