Customers arrive at a bank according to a Poisson process with
rate $\lambda$ per minute. Given that two customers arrived in the first 4
minutes, what is the probability that one arrived in the first minute and the other arrived in the last minute.
The setup for the problem is the following
$$P[N_1=1,N_3=1,N_3=1|N_4=2]=\frac{P[N_1=1,N_3=1,N_4=2]}{P[N_4=2]}$$
by bayes' theorem and conditionality which is equal to
$$\frac{P[N_1=1]P[N_2=0]P[N_1=1]}{P[N_4=2]}$$
I understand the inclusion of the term $P[N_2=0]$ due to the stationarity of increments of a poisson process but I don't understa…
