Suppose that in a certain small town it either rains or is sunny every day. A meteorologist notes that when it is sunny in this town, it will be sunny the next day with probability 0.6. If it rains, it will be sunny the next day with probability 0.3. Answer the following questions:1. **What is the probability that it will rain the next day if it was sunny the previous day?** - Answer: **0.4**2. **Suppose that the weatherman estimates there is a 70% chance of showers today. Find the probability vector representing this estimation where state 1 is sunny and state 2 is rainy.** - Probability Vector: \(\begin{bmatrix} 0.3 \\ 0.7 \end{bmatrix}\)3. **Based on the meteorologist’s estimation, what is the chance of rain tomorrow?** - Answer: **0.61**4. **Based on the meteorologist’s estimation, what is the chance of rain two days from now?** - Answer: **0.343**

Hello! As you have posted more than 3 sub parts, we are answering the first 3 sub-parts. In case you require the unanswered parts also, kindly re-post that parts separately. From the given information, A meteorologist notes that when it is sunny in this town, it will be sunny the next day with probability 0.6. If it rains, it will be sunny the next day with probability 0.3. The transition probability matrix is, P=0.60.40.30.7 1. The probability that it will rain the next day if it was sunny the previous day is, Prain|sunny=0.4 2. Suppose that the weatherman estimates there is a 70% chance of showers today. P(rainy)=70%=0.70 P(Sunny)=1-0.7=0.3 The probability vector representing this estimation where the state 1 is sunny 2 is rainy is, Here, the state 1 is sunny and state 2 is rainy: 0.30.7 3. The chance of rainy is, P(rainy)=0.70. I: Tomorrow rainy given that today is rainy P(rainy)=0.70*0.7=0.49 II: Tomorrow rainy given that today is sunny P(rainy)=0.70*0.4=0.28 Thus, the chance of rain tomorrow is, P(rain tomorrow)=I+II=0.49+0.28=0.77