Question

A certain town never has two sunny days in a row. Each day is classified as being either sunny, cloudy (but dry), or rainy. If it is sunny one day, then it is equally likely to be either cloudy or rainy the next day. If it is rainy or cloudy one day, then there is one chance in two that it will be the same the next day, and if it changes then it is equally likely to be either of the other two possibilities. In the long run, what proportion of days are sunny? What proportion are cloudy?

Short answer

In the long run, the proportion of sunny days is \( \frac{1}{3} \) and the proportion of cloudy days is also \( \frac{1}{3} \).

Step-by-step solution

Create the transition matrix

Create the transition matrix (P) using the provided probabilities. \[ P = \begin{bmatrix} 0 & 1/2 & 1/2 \\ 1/2 & 1/2 & 0 \\ 1/2 & 0 & 1/2 \\ \end{bmatrix} \] In this matrix, P[i][j] represents the probability of transitioning from state i to state j. For example, P[0][1] represents the probability of transitioning from Sunny (0) to Cloudy (1), which is 1/2.

Find the steady-state probabilities

Let X = [p_sunny, p_cloudy, p_rainy] be the steady-state probabilities of the states S, C, and R. Then we need to satisfy the following conditions: 1. X * P = X (steady-state condition) 2. p_sunny + p_cloudy + p_rainy = 1 (probabilities must add up to 1) Let's solve this system of equations to find the steady-state probabilities. Multiplying X with P: \[ [p_sunny, p_cloudy, p_rainy] \times \begin{bmatrix} 0 & 1/2 & 1/2 \\ 1/2 & 1/2 & 0 \\ 1/2 & 0 & 1/2 \\ \end{bmatrix} = [p_sunny, p_cloudy, p_rainy] \] We can now express the steady-state probabilities in the form of equations: p_sunny = (1/2)p_cloudy + (1/2)p_rainy p_cloudy = (1/2)p_sunny + (1/2)p_cloudy p_rainy = (1/2)p_sunny + (1/2)p_rainy p_sunny + p_cloudy + p_rainy = 1 Solve this system of equations to find the values of p_sunny, p_cloudy, and p_rainy.

Solve the system of equations

From the second and third equations above, we can deduce that p_cloudy = p_rainy. Now, we can rewrite the first equation as: p_sunny = (1/2)p_cloudy + (1/2)p_cloudy p_sunny = p_cloudy Since p_sunny + p_cloudy + p_rainy = 1, substitute the values we have found: p_sunny + p_sunny + p_sunny = 1 Now, solve for p_sunny: 3p_sunny = 1 p_sunny = 1/3 Since p_sunny = p_cloudy, we also have: p_cloudy = 1/3 Finally, we need to find p_rainy, knowing that the sum of all probabilities is 1: p_rainy = 1 - p_sunny - p_cloudy p_rainy = 1 - 1/3 - 1/3 p_rainy = 1/3

Determine the proportion of sunny and cloudy days

From the steady-state probabilities, we can deduce the proportion of sunny and cloudy days: In the long run: - Proportion of sunny days: p_sunny = 1/3 - Proportion of cloudy days: p_cloudy = 1/3 Therefore, in the long run, 1/3rd of the days will be sunny, and 1/3rd will be cloudy.