1. **Problem Statement:** We are given weather-dependent probabilities for rain based on the last two days' weather, and asked to model this as a Markov chain. We must identify the number of states and find the transition matrix. 2. **Why is this a Markov chain?** A Markov chain models a system where the next state depends only on the current state, not the entire history. Here, the future rain depends only on the weather of the last two days, so encoding these conditions as states yields the Markov property. 3. **Number of States:** Since the weather depends on the last two days, each state corresponds to a 2-day weather pattern. There are 2 possibilities for each day: rain (R) or no rain (N). Thus, total states = $$2^2 = 4$$. Define states: - State 0: Rain today (R), Rain yesterday (R) - State 1: Rain today (R), No rain yesterday (N) - State 2: No rain today (N), Rain yesterday (R) - State 3: No rain today (N), No rain yesterday (N) 4. **Constructing the Transition Matrix:** Each state transition moves forward one day; today becomes yesterday and tomorrow's rain status becomes today’s. For example, from state 0 (RR): - Probability it rains tomorrow = 0.7 - Probability no rain tomorrow = 0.3 Tomorrow’s state from 0 is either: - Rain tomorrow + rain today (which was yesterday) = state 0 (RR) with probability 0.7 - No rain tomorrow + rain today = state 2 (NR) with probability 0.3 Similarly from state 1 (RN): - Rain tomorrow probability 0.5 → next state 0 (RR) - No rain tomorrow 0.5 → next state 2 (NR) From state 2 (NR): - Rain tomorrow 0.4 → next state 1 (RN) - No rain tomorrow 0.6 → next state 3 (NN) From state 3 (NN): - Rain tomorrow 0.2 → next state 1 (RN) - No rain tomorrow 0.8 → next state 3 (NN) 5. **Transition Matrix** $$P$$ rows and columns correspond to states 0 to 3: $$ P = \begin{pmatrix} 0.7 & 0 & 0.3 & 0 \\ 0.5 & 0 & 0.5 & 0 \\ 0 & 0.4 & 0 & 0.6 \\ 0 & 0.2 & 0 & 0.8 \end{pmatrix} $$ - Entry $P_{ij}$ is the probability of transitioning from state $i$ to state $j$. 6. **Explanation:** The matrix correctly encodes all possible transitions and their probabilities as dictated by last two days’ rain and the given probabilities of rain tomorrow. **Final answer:** - The Markov chain has 4 states. - The transition matrix is given as above.