I've been given an assignment to model weekly passenger activity on three airlines: Delta, United and American.
The situation:
100 business travelers are followed during their weekly flights. No passengers leave or enter the system.
Delta: 75% of passengers who fly Delta on a given week choose to fly Delta the following week. 20% choose American the following week and 5% choose United.
United: 60% of passengers who fly United remain on United the following week. 20% choose to fly Delta and 20% choose to fly American the following week.
American: 40% of American passengers remain on American the following week. 20% switch to United and 40% switch to Delta.
The question:
What is the probability that no passenger flies Delta in week 10?
My work so far:
The initial distribution is even for all three weeks. I have set up my coefficient matrix $A$ with the numbers above, found my diagonal matrix $B$ such that $A=P^{-1}BP$ and written the equation to find the distribution for any given week $n$: $\pi_n = A^n\pi_0 = PB^nP^{-1}\pi_0$, where $\pi_0 = [1/3,1/3,1/3]$
Plugging into wolfram (which I'm allowed to do) I've gotten the distribution at week 10. I am not sure of the next step. My guess is that if I find the probability that one passenger flies Delta in week 10 I can subtract that probability from one and raise the result to the 100th power?