A supermarket you work part-time at has one express lane openfrom 5 to 6 PM on weekdays (Monday through Friday). This time ofthe day is usually the busiest since people tend to stop on theirway home from work to buy groceries. The number of items allowed inthe express lane is limited to 10 so that the average time toprocess an order is fairly constant at about 1 minute. The managerof the supermarket notices that there is frequently a long line ofpeople waiting and hears customers grumbling about the wait. Toimprove the situation he decides to open additional express lanesduring this time period. If he does, however, he will have to\"pull\" workers from other jobs around the store to serve ascashiers. Hence, he is reluctant to open more lanes thannecessary.
Knowing that you are a college student studying probability, yourmanager asks you to help him decide how many express lanes to open.His requirement is that there should be no more than one personwaiting in line 95% of the time.
With the task at hand, you set out to study the problem first. Youstart by counting the number of customer arrival in the expresslane on a Monday from 5 to 6pm. There are a total of 81 arrivals.You repeat the experiment on the following four days (Tuesdaythrough Friday) and note the total arrivals of 68, 72, 61 and 66customers, respectively.

1) What is the average number of customer arrivals at the expresslane from 5 to 6pm on weekdays?

2) Assume the customer arrivals at the express lane from 5 to6pm on weekdays can be modeled by a Poisson random variable, whatis the PMF for the number of customers arrived during a one-minuteinterval in this period?

3) What is the probability of two or fewer customers arriving atthe one express lane during a oneminute interval in this period?Does it satisfy the manager’s requirement of no more than oneperson waiting in line 95% of the time?

4) If your answer to the previous question is no, how manyexpress lanes should the manager open in order to satisfy hisrequirement? You can assume that the arriving customer is equallylikely to join any of the express lane if there are more than oneexpress lanes. Also you can assume the lanes are independent, butall lanes must satisfy the manager’s requirement.