(b) Patients arrive at a hospital accident and emergency department at random at a rate of 6 per hour. Find the probability that, during any 90 min. period, the number of patients arriving at the hospital accident and emergency department is (i) exactly 7 (ii) A patient arrived at 11:30 a.m., then find probability that the next patient arrives before 11:45 a.m.

(i) Let X be a random variable represents the number of patients arrived in 1.5 hours, then X ~ Pois(6*1.5)=Pois(9)

$P(X=7)={\frac {9^7} {7!}}*e^{-9}\approx0.117$

(ii) Let X be a random variable represents the number of patients arrived in 15 minutes hours, then X ~ Pois(6*0.25)=Pois(1.5)

Next patient arrives before 11:45 a.m means that at least one patient will arrive in the next 15 minutes

$P(X≥1)=1-P(X<1)=1-P(X=0)=1-{\frac {1.5^0} {0!}}*e^{-1.5}\approx 0.777$