Question

Weather experts estimate the probability of rain any days as
0.6 if it rained the previous day
0.3 if it did not rain the previous day
Fine the probability that a dry day is followed by
(a) Two more dry days
(b) Two wet days
(c) A wet day and then a dry day

Answer 1

To solve this problem, we can represent the weather as a Markov chain with two states: "Dry" (D) and "Wet" (W). We are provided with the following probabilities:

- If it rained the previous day, the probability of rain the next day is 0.6, thus the probability of a dry day following a wet day is:
- P(D | W) = 1 - 0.6 = 0.4

- If it did not rain the previous day, the probability of rain the next day is 0.3, thus the probability of a dry day following a dry day is:
- P(D | D) = 1 - 0.3 = 0.7

This leads us to the following transition probabilities:
- P(W | W) = 0.6 (wet today, wet tomorrow)
- P(D | W) = 0.4 (wet today, dry tomorrow)
- P(W | D) = 0.3 (dry today, wet tomorrow)
- P(D | D) = 0.7 (dry today, dry tomorrow)

Now, we want to find the probabilities for the different scenarios following a dry day:

### (a) The probability that a dry day is followed by two more dry days:
We want to find P(D | D) x P(D | D):
- The probability of a dry day followed by another dry day is P(D | D) = 0.7.
- The probability of the next day (after the second dry day) also being dry is again P(D | D) = 0.7.

So, the total probability is:
\[
P(\text{D, D, D}) = P(D | D) \times P(D | D) = 0.7 \times 0.7 = 0.49.
\]

### (b) The probability that a dry day is followed by two wet days:
We want to find P(W | D) x P(W | W):
- The probability of a dry day followed by a wet day is P(W | D) = 0.3.
- The probability of the next day (after the first wet day) also being wet is P(W | W) = 0.6.

So, the total probability is:
\[
P(\text{D, W, W}) = P(W | D) \times P(W | W) = 0.3 \times 0.6 = 0.18.
\]

### (c) The probability that a dry day is followed by a wet day and then a dry day:
We want to find P(W | D) x P(D | W):
- The probability of a dry day followed by a wet day is P(W | D) = 0.3.
- The probability of the next day (after the wet day) being dry is P(D | W) = 0.4.

So, the total probability is:
\[
P(\text{D, W, D}) = P(W | D) \times P(D | W) = 0.3 \times 0.4 = 0.12.
\]

### Summary of Probabilities:
(a) The probability that a dry day is followed by two more dry days: **0.49**
(b) The probability that a dry day is followed by two wet days: **0.18**
(c) The probability that a dry day is followed by a wet day and then a dry day: **0.12**