Question

An extremely simple (and surely unreliable) weather prediction model would be one where days are of two types: sunny or rainy. 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. Model this as a Markov chain. If Sunday is sunny, what is the probability that Tuesday (two days later) is also sunny

Answer 1

The probability that if Sunday is sunny, then Tuesday is also sunny is 0.86.

Explanation:

Let us denote the events as follows:

Event 1: a sunny day

Event 2: a rainy day

From the provided data we know that the transition probability matrix is:

`\left\begin{array}{ccc}1&\ \ \ \ 2\end{array}\right`

`\text{P}=\left\begin{array}{c}1&2\end{array}\right`
`\left[\begin{array}{cc}0.90&0.10\\0.50&0.50\end{array}\right]`

In this case we need to compute that if Sunday is sunny, what is the probability that Tuesday is also sunny.

This implies that we need to compute the value of P₁₁².

Compute the value of P² as follows:

`P^(2)=P\cdot P`

`=\left[\begin{array}{cc}0.90&0.10\\0.50&0.50\end{array}\right]\cdot \left[\begin{array}{cc}0.90&0.10\\0.50&0.50\end{array}\right]\\\\=\left[\begin{array}{cc}0.86&0.14\\0.70&0.30\end{array}\right]`

The value of P₁₁² is 0.86.

Thus, the probability that if Sunday is sunny, then Tuesday is also sunny is 0.86.