Consider this ismple meterological model. We are interested in modelling rainfal

Question

Consider this ismple meterological model. We are interested in modelling rainfall occurrence during a series of consecutive days. Let Ri denote the event that it rains on day i. Suppose that P(Ri | Ri-1) = a and P(Ric | Ri-1c) = b. Also assume that only toda'ys weather is relevant to predicitng tomorrow's weather. This is called Markov property.

a) The probability of rain today is p. What is the probability of rain tomorrow? Hint: use the Law of Total Probability.

b) What is the probability of rain the day after tomorrow?

Explanation / Answer

a) The probability of rain today is p, this means that:

P( rain today ) = p and therefore P( no rain today ) = 1 - p

Now we are given P( rain tomorrow | rain today ) = a and

Also we are given that P( no rain tomorrow | no rain today ) = b that means P( rain tomorrow | no rain today ) = 1 - b

Now using the Law of total probability we get:

P( rain tomorrow ) = P( rain tomorrow | rain today )P( rain today ) + P( rain tomorrow | no rain today )P( no rain today )

P( rain tomorrow ) = ap + (1-b)(1-p) = ap + 1 -b -p + bp

Therefore the probability of rain tomorrow is given as: 1 + ap + bp - b - p

b) Probability of rain day after tomorrow ?

P( rain day after tomorrow | rain tomorrow ) = a and

P( rain day after tomorrow | no rain tomorrow ) = 1- b

Therefore, again using the law of total probability we get:

P( rain day after tomorrow ) = P( rain day after tomorrow | rain tomorrow ) P( rain tomorrow ) +
P( rain day after tomorrow | no rain tomorrow ) P( no rain tomorrow )

P( rain day after tomorrow ) = a*(1 + ap + bp - b - p) + (1-b) *( b + p - ap- bp )

P( rain day after tomorrow ) = a + a2p + abp - ab - ap + b + p -ap - bp - b2 - bp + abp + b2p

P( rain day after tomorrow ) = a + b + a2p + b2p + 2abp + p - ab - 2ap - 2bp

This is the required probability here.

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