Martin commutes to work. Each day he either drives his car, takes the bus, or takes a taxi.
He never drives his car two days in a row.
There is a 19% chance that he will drive if he did NOT drive the previous day.
He is always twice as likely to take the bus as he is to take a taxi. (Example: If the probability that he takes a taxi is 1/4, then the probability that he takes a bus is twice that: 2(1/4) = 2/4 = 1/2.)
This gives you enough information to set up the transition matrix. (Hint: When creating transition matrices it is important not to round any of the numbers because when you start multiplying matrices the round-off error will get compounded. To avoid this, use fractions instead of approximated decimals. For example, you would put a fraction such as 5/6 in your matrix as "5/6" instead of using the decimal approximation "0.83333333".)
Use the matrix to answer the following questions. (Give your answers correct to three decimal places.)
(a) If Martin drives on Monday, what is the probability he will take the bus on Wednesday?
(b) If Martin drives on Monday, what is the probability he will take a taxi on Thursday?
If Martin drives on Monday, what is the probability he will drive on Thursday?
(c) In the long run, what fraction of the time does he drive?
In the long run, what fraction of the time does he take the bus?
In the long run, what fraction of the time does he take a taxi?