assuming every day's weather has only two options: rainy or sunny.

the probability that in certain day the weather is the same as the day before is $\displaystyle p$. the probability of change in weather is $\displaystyle q=1-p$.

now i will define $\displaystyle (x_{[n]})_{n>=0}$ as a discrete time stochastic process as: $\displaystyle x_{[n]}=1$ if the n'th day was rainy and $\displaystyle x_{[n]}=0$ if the n'th day was sunny.

the question is: given that today, n=0, is a rainy day, find the probability to rain in the n'th day. clue: you can organize the data in a matrix.