# Given a transition matrix P for weather conditions (modeled as either rainy or sunny), is $P^n$ the n-Step Transition Probabilities for day n+1?

wiki uses this example to illustrate Markov chains.

The probabilities of weather conditions (modeled as either rainy or sunny), given the weather on the preceding day, can be represented by a transition matrix:

$${\displaystyle P={\begin{bmatrix}0.9&0.1\\0.5&0.5\end{bmatrix}}}$$

The matrix P represents the weather model in which a sunny day is 90% likely to be followed by another sunny day, and a rainy day is 50% likely to be followed by another rainy day. The columns can be labelled "sunny" and "rainy", and the rows can be labelled in the same order.

The weather on day 1 is known to be sunny. This is represented by a vector in which the "sunny" entry is 100%, and the "rainy" entry is 0%:

$${\displaystyle \mathbf {x} ^{(0)}={\begin{bmatrix}1&0\end{bmatrix}}}$$

for day n + 1(Note: the original value on wiki is n, which seems to be incorrect)

$${\mathbf {x}}^{{(n)}}={\mathbf {x}}^{{(0)}}P^{n}$$

The superscript (n) is an index, and not an exponent.

is $$P^n$$ here the n-Step Transition Probabilities?

Yes, $$P^n$$ represents the $$n$$-step transition probability matrix. Here is one way to show that. According to Chapman-Kolmogorov equations, the $$(n+m)$$-step transition probabilities can be expressed as the sum of products of $$n$$-step and $$m$$-step probabilities,as follows:
$$p_{ij}^{(n+m)}=\sum_{l}p_{il}^{(n)}p_{lj}^{(m)}$$
or, using matrix notation as $$P^{(n+m)}$$=$$P^{(n)}P^{(m)}$$. Substituting in the last expression $$n-1$$ for $$n$$ and $$1$$ for $$m$$ we get $$P^{(n-1+1)}$$=$$P^{(n-1)}P^1$$, that is $$P^{(n)}$$=$$P^{(n-1)}P^1$$. Applying recursion to the right side, we arrive at
$$P^{(n)}=P^n$$
in other words, the $$n$$-step transition matrix is the n-th power of the 1-step transition matrix.