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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

1 answer

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