There are slides regrading to urn model
I have two questions
1.if a Species A dies and a Species A is born, the original text says the probability is 0.4*0.4, but since a Species A has died , only one Species A is left, shouldn't the probability for a new bornspecies A be (2-1)/(5-1)=1/4=0.25?
2.I don't understand what the row vector means? since it has 6 components, does it mean it has 6 individuals or 6 species?
On
$$P = \begin{bmatrix}\\Row&5A0B & 4A1B & 3A2B & 2A3B & 1A4B & 0A5B \\ 5A0B & 1 & 0 & 0 &0&0 &0\\4A1B& 0.16 & 0.68 & 0.16 & 0 & 0 &0\\3A2B &0&0.24&0.52&0.24&0&0\\2A3B&0&0&0.24&0.52&0.24&0\\1A4B&0&0&0&0.16&0.68&0.16\\0A5B&0&0&0&0&0&1\end{bmatrix}$$
The top row indicates the different states and they are six of them. Your second question mark states that the equilibrium states could be that either species A survives or B survives ultimately and they are called the absorbing states.
The slides are a bit unclear on your first point, but from the formulation "since $3$ of the $5$ individuals in the system are Species A" we can infer that birth and death are considered to occur simultaneously, and the individual selected for death has the same chance of giving birth as all others.
On your second question: There are $6$ different states because there can be anywhere from $0$ to $5$ individuals of Species A; that's $6$ different cases.