## Stochastic Process - Creating a Probability Transition Matrix

1. The problem statement, all variables and given/known data

The total population size is N = 5, of which some are diseased and the rest are healthy. During any single period of time, two people are selected at random from the population and assumed to interact. The selection is such that an encounter between any pair of individuals is independent of any other pair. If one of the persons is diseased and the other isn't, the with probability 0.1 the disease is transmitted to the healthy person. Otherwise, no disease is transmitted. Let denote the number of diseased people in the population at the end of the nth period. Find the transition probability matrix.
2. Relevant equations

3. The attempt at a solution

I did the TPM for two people from the population:

H ~ Healthy Person D ~ Diseased person

$$\begin{bmatrix}0.9 & 0.1\\ 0 & 1\end{bmatrix}$$

The columns are (H,D) and the rows are (H,D)'

But I'm not sure how to account for the N=5 population
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