An airline runs a frequent flyer scheme with four classes of member: in ascending order Ordinary. Bronze, Silver and Gold. Members receive benefits according to their class. Members who book two or more flights in a given calendar year move up one class for the following year (or remain Gold members), members who book exactly one flight in a given calendar year stay at the same class, and members who book no flights in a given calendar year move down one class (or remain Ordinary members). Let the proportions of members booking 0, 1 and 2+ flights in a given year be pop and P2+ respectively. (1) (a) Explain how this scheme can be modelled as a Markov chain. (b) Explain why there must be a unique stationary distribution for the proportion of members in each class. (i) Write down the transition matrix of the process. The airline's research has shown that in any given year, 40% of members book no flights, 40% book exactly one flight, and 20% book two or more flights. (ii) Calculate the stationary probability distribution. The cost of running the scheme per member per year is as follows: Ordinary members 0 Bronze members 10 Silver members 20 Gold members 30 The airline makes a profit of 10 per passenger for every flight before taking into account costs associated with the frequent flyer scheme. (iv) Assess whether the airline makes a profit on the members of the scheme