The number of patients arriving at the emergency room (ER) at a large city hospital is a discrete random variable that can be modelled with a Poisson distribution with a mean on 10 patients per hour. A random variable Yi is defined to represent the amount of time passing between the arrival time of Patient i and arrival time of patient i-1, where i=1,2,3,.... Specifically,
Yi=Xi-Xi-1 where Xi=arrival time of patient i, i=2,3,...
A random variable Z is defined as the total amount of time passing until the fourth patient arrives at the ER in specific one hour period. What is the probability that at least 10 minutes to at most 20 minutes pass from the start of the hour until the fourth patient arrives at the ER?