Exercise 1 (10 pts). The toll booth operator cannot take a break as long as there are cars in her queue. The times between arrivals of cars are independent with mean 5 minutes and finite variance. It takes her on average 2 minutes to process each customer and the time to process a car has finite variance. These service times are independent for different customers and independent of the arrival times of the cars. Let pn be the probability that the toll booth operator has had to serve n customers in a row without a break. Show that Pn - 0 as n -+ 00. Hint: Denote qn+1 the probability that the n + 1th car arrives before the first n cars have been processed. Show that Pn+1