Question

Suppose that the chance of rain tomorrow depends on previous weather conditions only through whether or not it is raining today and not on past weather conditions. Suppose also that if it rains today, then it will rain tomorrow with probability 0.6, and if it does not rain today, then it will rain tomorrow with probability 0.3, then

a. Calculate the probability that it will rain four days from today given that it is raining today.

b. What is the limiting probability of rain.

Answer 1

a.

The problem can be modeled as Markov chain with states R {Rain} and NoRain {N}.

The transition probability from state R to state R is 0.6. So, The transition probability from state R to state N is 1 - 0.6 = 0.4

The transition probability from state N to state R is 0.3. So, The transition probability from state N to state N is 1 - 0.3 = 0.7

The one step transition probability matrix is,

The transition probability matrix after four days is,

The probability that it will rain four days from today given that it is raining today is 0.4332 (Transition from state R to state R in P4)

b.

Let be the limiting distribution of transition probability matrix.

Then,

and r + n = 1

which gives,

0.6r + 0.3n = r => 0.4r = 0.3n => n = 4r/3

0.4r + 0.7n = n

r + n = 1 => r + 4r/3 = 1

=> 7r/3 = 1

=> r = 3/7

The limiting probability of rain is 3/7 = 0.4285714