Poisson to Binomial Distribution Proof?

Question

Q:Let {N(t) : t ≥ 0} be a Poisson process. For s = t/3, show that the conditional distribution of N(s) given N(t) = n is binomial with parameters n and p = 1/3. Also, find the conditional distribution of N(t) given N(s) = k.

My attempt:

We know the following,

1. N(s) ~Poisson(L.s)
2. N(t) ~Poisson(L.t)
3. N(t) - N(s) ~Poisson(L.(t-s)) -> N(t)|N(s)~Poisson(L.(t-s))

furthermore we need N(s)|N(t) so we can apply Bayes rule,

P(A|B)=P(B|A).P(A)/P(B) =>

P(N(s))|N(t) = {Poisson(L.s)+Poisson(L.(t-s))}.Poisson(L.s)/Poisson(L.t)

This is where I am stuck. I am not sure if even my approach is right. Any help would be great.