With over two hundred other postings, you should understand how this forum works. So please either post some of your own work on this problem or explain what you do not understand about the question.
Whenever a commuter has travelled by car, then the next day there is probability 1/3 that he takes the car and probability 2/3 that he takes the train. Whenever he has travelled by train, then the next day there is probability 1/2 that he takes the car and probability 1/2 that he trakes the train.
Write out the transition matrix for this Markov chain.
Help is greatly appreciated!