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 60% of chance 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, how frequent are we observing a day for which it is rainy and the next two days are also rainy

Answer 1

In the long run if one day is rainy there is frequency of 2/5 that other two days will also be rainy.

Explanation:

A matrix is formed to classify the weather conditions. It is given that if one day is sunny then there is likely chance that the next day will be cloudy, but if one day is rainy then there is 60% chance that next day will be same. To identify the possibility of next two days we create probability matrix;

Probability (P) =
`\left[\begin{array}{ccc}0&(1)/(2) &(1)/(2) \\(1)/(4) &(1)/(2) &(1)/(4) \\(1)/(4) &(1)/(4) &(1)/(2) \end{array}\right]`

After solving for probability we get a fraction of 2/5