The problem is the following:
An individual has three umbrellas, some at her office, and
some at home. If she is leaving home in the morning (or leaving work at night) and it
is raining, she will take an umbrella, if one is there. Otherwise, she gets wet. Assume
that independent of the past, it rains on each trip with probability 0.2. To formulate
a Markov chain, let be the number of umbrellas at her current location.
(a) Find the transition probabilities for this Markov chain.
(b) Calculate the limiting fraction of time she gets wet.
For part a) I have written the following matrix:
$\begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0.8 & 0.2 \\ 0 & 0.8 & 0.2 & 0 \\ 0.8 & 0.2 & 0 & 0 \end{bmatrix}$
where the rows and columns represent the number of umbrellas in her particular location (we are transitioning from row i to column j).
Though I think my solution to part a) might be right, I don't know how to approach part b). Can someone give me a hand? I would appreciate it.
2026-02-24 05:06:33.1771909593
Find transition matrix for Markov Chain
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