Ok, so my question is pretty simple, the question states:
A spider web is only big enough to hold 2 flies at a time. Assuming that the flies fly into the web independently:
-The probability that no flies will fly into her web on any given day is $0.5$.
-The probability that exactly one fly will fly into her web on any given day is $0.3$.
-The probability that two or more flies will fly into her web on any given day is $0.2$.
It also states that if a fly flies into the web when the web is full, it will bounce off and escape. Every morning the spider checks the web and will always eat a flies if there is one available, but can only eat 1 a day, leaving any left for the next day.
So, my transition matrix for this is:
$$M = \begin{bmatrix}0.5 & 0.3 & 0.2\\0.5 & 0.3 & 0.2\\0 & 0.5 & 0.5 \end{bmatrix}$$
The working after this is pretty simple, I'm just not sure if I've done the matrix correctly, any help is appreciated.
Yes, you are correct and the below table will explain.