# A simple probability problem involving the law of total probability

## Homework Statement

This problem introduces a simple meteorological model, more complicated versions of which have been proposed in the meteorological literature. Consider a sequence of days and let Ri denote the event that it rains on day i . Suppose that P(Ri | Ri−1) = α and P(Rci | Rci−1c) = β.

Note that Rci is the complement of Ri.

Suppose further that only today’s weather is relevant to predicting tomorrow’s; that is, P(Ri| Ri−1 ∩ Ri−2 ∩···∩ R0) = P(Ri| Ri−1).

a. If the probability of rain today is p, what is the probability of rain tomorrow?
b. What is the probability of rain the day after tomorrow?
c. What is the probability of rain n days fromnow?What happens as n approaches inﬁnity?

## The Attempt at a Solution

Part (a) is pretty easy, I just apply the total probability formula and it should give me the answer.
P(R1) = P(R1 | R0)*P(R0) + P(R1 | Rc0)*P(Rc0)
P(R1) = α*p + (1-B)*(1-p) = p(α-1+B) + (1-B)

Part (b) is tedious if I just apply the total probability formula again and use P(R1) I found in part (a).
P(R2) = p(α-1+B)^2 + (1-B)(α+B)

Part (c), since there appears to be a repetitive pattern in the answer in (b) and (a), I solved for P(R3) to try and confirm this, and got
P(R3) = p(α-1+B)^3 + (1-B)[(α+B)(α-1) + 1]

so I'm thinking that my method in approaching the last part in inaccurate because I can't see the pattern and I haven't used the other piece of information that was given.

Suppose further that only today’s weather is relevant to predicting tomorrow’s; that is, P(Ri| Ri−1 ∩ Ri−2 ∩···∩ R0) = P(Ri| Ri−1).

Any hint in the right direction is appreciated.
Thanks.

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Question 3) is slightly different from questions 1) and 2). There, they asked you the probability of raining on one fixed day ($$pR_{1}$$ for question 1) and $$pR_{2}$$ for question 2)).
$$p(\bigcap^{n}_{i=1}R_{i})$$