CT4 September 2011 Question 11 (ii)

"An actuary walks from his house to the office each morning, and walks back again each evening. He owns two umbrellas. If it is raining at the time he sets off, and one or both of his umbrellas is available, he takes an umbrella with him. However if it is not raining at the time he sets off he always forgets to take an umbrella.
Assume that the probability of it raining when he sets off on any particular journey is a constant p, independent of other journeys."

Part (ii) of the question asks to derive the transition matrix for the number of umbrellas at the actuary’s house before he leaves each morning, based on the number before he leaves the previous morning.

This was my solution, and I want to understand why it's incorrect?

The paper solution is below: