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 lst Januaryis sunny is 0.8. Find the probability that (a) 2nd January is sunny. (b) 3rd January is rainy. (c) 3rd January is sunny given that Ist January is sunny, (d) 1st January is rainy given that 3rd January is sunny.

(a) The probability that 2nd January is sunny:

We know that the probability that 1st January is sunny is 0.8. Given that a sunny day is followed by a sunny day with a probability of 0.75, we can multiply these two probabilities together to find the probability that 2nd January is sunny.

0.8 (probability that 1st January is sunny) * 0.75 (probability that a sunny day is followed by a sunny day) = 0.6

So, the probability that 2nd January is sunny is 0.6.

(b) The probability that 3rd January is rainy:

To find this, we need to consider two scenarios: (1) 2nd January is sunny and followed by a rainy day, and (2) 2nd January is rainy and followed by a rainy day.

For the first scenario, we already know that the probability that 2nd January is sunny is 0.6. The probability that a sunny day is followed by a rainy day is 1 - 0.75 = 0.25. So, the probability of the first scenario is 0.6 * 0.25 = 0.15.

For the second scenario, the probability that 2nd January is rainy is 1 - 0.6 = 0.4. The probability that a rainy day is followed by a rainy day is 0.7. So, the probability of the second scenario is 0.4 * 0.7 = 0.28.

Adding these two probabilities together, we get 0.15 + 0.28 = 0.43.

So, the probability that 3rd January is rainy is 0.43.

(c) The probability that 3rd January is sunny given that 1st January is sunny:

This is the same as the probability that 2nd January is sunny (which we already calculated as 0.6) multiplied by the probability that a sunny day is followed by a sunny day (0.75).

0.6 * 0.75 = 0.45

So, the probability that 3rd January is sunny given that 1st January is sunny is 0.45.

(d) The probability that 1st January is rainy given that 3rd January is sunny:

This is a more complex problem that requires the use of Bayes' theorem. However, with the information given in the problem, it's not possible to calculate this probability. We would need additional information about the overall likelihood of sunny and rainy days.