Suppose the weather of a day is cither sunny or rainy. The probability that a sunny day is followed by a sunnyday is 0.75. The probability that a rainy day is followed by a rainy day is 0.7. The probability that 1st Januaryis sunny is 0.8. Find the probability that 1st January is rainy given that 3rd January is sunny.

To solve this problem, we need to use the concept of conditional probability and the given probabilities.

Step 1: Define the events Let's define the following events: S1: 1st January is sunny S3: 3rd January is sunny R1: 1st January is rainy

Step 2: Use the given probabilities We know that: P(S1) = 0.8 (probability that 1st January is sunny) P(S1->S2) = 0.75 (probability that a sunny day is followed by a sunny day) P(R1->R2) = 0.7 (probability that a rainy day is followed by a rainy day)

Step 3: Calculate the complementary probabilities We can calculate the complementary probabilities as follows: P(R1) = 1 - P(S1) = 1 - 0.8 = 0.2 (probability that 1st January is rainy) P(S1->R2) = 1 - P(S1->S2) = 1 - 0.75 = 0.25 (probability that a sunny day is followed by a rainy day) P(R1->S2) = 1 - P(R1->R2) = 1 - 0.7 = 0.3 (probability that a rainy day is followed by a sunny day)

Step 4: Calculate the joint probabilities We can calculate the joint probabilities as follows: P(S1 and S2) = P(S1) * P(S1->S2) = 0.8 * 0.75 = 0.6 P(S1 and R2) = P(S1) * P(S1->R2) = 0.8 * 0.25 = 0.2 P(R1 and S2) = P(R1) * P(R1->S2) = 0.2 * 0.3 = 0.06 P(R1 and R2) = P(R1) * P(R1->R2) = 0.2 * 0.7 = 0.14

Step 5: Calculate the conditional probability We want to find P(R1|S3). Since S3 can be either after S2 or R2, we can write: P(R1|S3) = P(R1 and S2) / (P(S1 and S2) + P(R1 and S2)) = 0.06 / (0.6 + 0.06) = 0.06 / 0.66 = 0.0909

So, the probability that 1st January is rainy given that 3rd January is sunny is approximately 0.0909 or 9.09%.