Probability of rain on given day

Question

The probability of rain on a given day is $\frac{5}{17}$ if there was raining on the previous day, and $\frac{4}{13}$ if it wasn't on the previous day. What is the probability that there will be raining in $31$ days if it rains today?

Answer 1

We can model this situation with a discrete-time Markov chain with the two following states:

• State $1$ represents the state in which there was rain on the previous day.

• State $2$ represents the state in which there was not rain on the previous day.

The transition probabilities are given by $p_{11} = 9/13$, $p_{12} = 4/13$, $p_{21} = 12/17$ and $p_{22} = 5/17$

Now we can reformulate our problem as follows: "Given that we are currently in State $1$, what's the probability that we are in State $1$ after $30$ transitions?" This is given by $p_{1, 1}^{(30)}$. Exponentiating the transition matrix, we conclude this value is given by $\boxed{0.70}$