1. **State the problem:** We have a weather system where the probability of rain tomorrow depends on whether it rained today and yesterday. We are asked to show this can be analyzed using a Markov chain, determine how many states are needed, and find the transition matrix. 2. **Identify states:** Since the probability depends on the last two days' weather, we need to consider pairs of weather states for consecutive days. Possible states representing last two days are: - RR (rained both days) - RN (rained yesterday but not today) - NR (did not rain yesterday but rained today) - NN (did not rain either day) Thus, there are $4$ states. 3. **Define the state transitions:** From each state, we look at the probability of rain tomorrow, updating the two-day window accordingly. - From RR: next day rain with probability $0.7$, no rain with probability $0.3$. - Rain tomorrow $ ightarrow$ new state RR - No rain tomorrow $ ightarrow$ new state RN - From RN: rain tomorrow $0.5$, no rain $0.5$ - Rain tomorrow $ ightarrow$ NR - No rain tomorrow $ ightarrow$ NN - From NR: rain tomorrow $0.4$, no rain $0.6$ - Rain tomorrow $ ightarrow$ RR - No rain tomorrow $ ightarrow$ RN - From NN: rain tomorrow $0.2$, no rain $0.8$ - Rain tomorrow $ ightarrow$ NR - No rain tomorrow $ ightarrow$ NN 4. **Write the transition matrix:** Ordering states as [RR, RN, NR, NN], rows represent current states, columns represent next states. $$ P = \begin{bmatrix} 0.7 & 0.3 & 0 & 0 \\ 0 & 0 & 0.5 & 0.5 \\ 0.4 & 0.6 & 0 & 0 \\ 0 & 0 & 0.2 & 0.8 \end{bmatrix} $$ 5. **Explain:** This setup defines a Markov chain with $4$ states because the future (tomorrow's rain) depends only on the current state (last two days), satisfying the Markov property. The transition matrix gives the probabilities for moving between states. **Final answer:** - Number of states needed: $4$ - Transition matrix: $$ \begin{bmatrix} 0.7 & 0.3 & 0 & 0 \\ 0 & 0 & 0.5 & 0.5 \\ 0.4 & 0.6 & 0 & 0 \\ 0 & 0 & 0.2 & 0.8 \end{bmatrix} $$