This problem introduces a simple meteorological model,
Chapter , Problem 54(choose chapter or problem)
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 \(R_i\) denote the event that it rains on day i. Suppose that \(P(R_i|R_{i-1})=\alpha\) and \(P(R^c_i|R^c_{i−1})=\beta\). Suppose further that only today's weather is relevant to predicting tomorrow’s; that is, \(P(R_i|R_{i-1} \cap R_{i-2} \cap \cdots \cap R_0) = P(R_i|R_{i-1})\).
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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