This exercise is part of course in Probability and Statistics at Jio Institute.
Problem Definition:
Consider the following analysis of bus ridership, which (in more complex form) could be used by the bus company/agency to plan the number of buses, frequency of stops and so on. Again, in order to keep things easy, it will be quite oversimplified, but the principles will be clear.
Here is the model: • At each stop, each passenger alights from the bus, independently of the actions of others, with probability 0.2 each. • Either 0, 1 or 2 new passengers get on the bus, with probabilities 0.5, 0.4 and 0.1, respectively. Passengers at successive stops act independently. • Assume the bus is so large that it never becomes full, so the new passengers can always board. • Suppose the bus is empty when it arrives at its first stop.
Here and throughout, it will be greatly helpful to first name the quantities or events involved. Let Li denote the number of passengers on the bus as it leaves its ith stop, i = 1, 2, 3, . . . Let Bi denote the number of new passengers who board the bus at the ith stop.
Find the probability that after visiting the tenth stop, the bus is empty.