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This problem introduces a simple meteorological model more

Question: 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(R_i | R_i-1) = α and P(Rc_i| Rc_i-1) = β. Suppose further that only today's weather is relevant to predicting tomorrow's; that is, P(R_i| R_i-1 ∩ R_i-2 ∩···∩ R₀) = P(R_i | R_i-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 from now? What happens as n approaches infinity?

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