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.the answer is 0.146,why?

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

Let's denote: S1: 1st January is sunny S3: 3rd January is sunny R1: 1st January is rainy

We are asked to find P(R1|S3), the probability that 1st January is rainy given that 3rd January is sunny.

According to the total probability theorem, we can express P(S3) as the sum of the probability that 3rd January is sunny given 1st January was sunny times the probability 1st January was sunny, and the probability that 3rd January is sunny given 1st January was rainy times the probability 1st January was rainy:

P(S3) = P(S3|S1)P(S1) + P(S3|R1)P(R1)

We know that P(S1) = 0.8, so P(R1) = 1 - P(S1) = 0.2.

We also know that the probability that a sunny day is followed by a sunny day is 0.75, so P(S3|S1) = 0.75^2 = 0.5625 (since we have two sunny days following each other).

The probability that a rainy day is followed by a sunny day is 1 - 0.7 = 0.3, so P(S3|R1) = 0.3^2 = 0.09 (since we have a rainy day followed by a sunny day).

Substituting these values into the equation for P(S3) gives:

P(S3) = 0.56250.8 + 0.090.2 = 0.45 + 0.018 = 0.468

Now, we can find P(R1|S3) using Bayes' theorem:

P(R1|S3) = P(S3|R1)P(R1) / P(S3) = 0.09*0.2 / 0.468 = 0.0384 / 0.468 = 0.08205 ≈ 0.082

So, the probability that 1st January is rainy given that 3rd January is sunny is approximately 0.082, not 0.146. There might be a mistake in the problem statement or in the provided answer.