Suppose the weather of a day is either 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 total probability theorem. Here are the steps:

1. Let's denote the events as follows:

   • S1: The event that 1st January is sunny.
   • S3: The event that 3rd January is sunny.
   • R1: The event that 1st January is rainy.
   • R3: The event that 3rd January is rainy.

2. From the problem, we know the following probabilities:

   • P(S1) = 0.8 (The probability that 1st January is sunny)
   • P(S1->S3) = 0.75 (The probability that a sunny day is followed by a sunny day)
   • P(R1->R3) = 0.7 (The probability that a rainy day is followed by a rainy day)

3. We also know that the sum of the probabilities of all possible outcomes is 1. So, we can find the probability that 1st January is rainy as follows:

   • P(R1) = 1 - P(S1) = 1 - 0.8 = 0.2

4. Now, we need to find the probability that 3rd January is sunny. We can use the total probability theorem for this:

   • P(S3) = P(S1->S3) * P(S1) + (1 - P(R1->R3)) * P(R1)
   • P(S3) = 0.75 * 0.8 + (1 - 0.7) * 0.2 = 0.6 + 0.06 = 0.66

5. Finally, we can find the probability that 1st January is rainy given that 3rd January is sunny using Bayes' theorem:

   • P(R1|S3) = [(1 - P(S1->S3)) * P(R1)] / P(S3)
   • P(R1|S3) = [(1 - 0.75) * 0.2] / 0.66 = 0.075 / 0.66 = 0.1136

So, the probability that 1st January is rainy given that 3rd January is sunny is approximately 0.1136 or 11.36%.