John travels to work using one of three modes of transport – bicycle, car, or train. The mode of transport he uses on a given day, varies according to the following rules:
If John cycles today, then the probability of him cycling tomorrow is 0.7 and the probability of him travelling by car is 0.2.
If John travels by car today, then the probability of him cycling tomorrow is 0.3 and the probability of him travelleing by train is 0.3
. Finally, if John travels by train today, then the probability of him cycling tomorrow is 0.2 and the probability of him travelling by car is 0.4.
Assuming the above system can be described using a Markov Chain, answer the following.
(a) Construct a transition diagram, clearly labelling the probabilities, to represent this situation. (b) Construct a transition matrix to represent this situation
(c) If John cycles today, what is the probability that he will travel by car in 4 days’ time?
d) Determine the long-term probability for each scenario