Suppose that the bank customers arrive randomly and independently on an average of 3.2 customers every 4 minutes. What is the probability that: a. Exactly two customers arrive in every 4 minutes? b. Exactly two customers will arrive in every 8 minutes interval? c. One or more customers will arrive in every 12 minutes?

a) Let X be a random variable represents the number of customers arriving in 4 minutes, then X~Pois(3.2)

$P(X=2)={\frac {3.2^2} {2!}}*e^{-3.2}\approx0.209$

b) Let X be a random variable represents the number of customers arriving in 8 minutes, then X~Pois(2*3.2)=Pois(6.4)

$P(X=2)={\frac {6.4^2} {2!}}*e^{-6.4}\approx0.034$

b) Let X be a random variable represents the number of customers arriving in 12 minutes, then X~Pois(3*3.2)=Pois(9.6)

$P(X≥1)=1-P(X<1)=1-P(X=0)=1-{\frac {9.6^0} {0!}}*e^{-9.6}\approx1$