As a simplified model for weather forecasting, suppose that the weather (either

Question

As a simplified model for weather forecasting, suppose that the weather (either wet or dry) tomorrow will be the same as the weather today with probability p. Suppose that the weather is dry on January 1st and let Pn be the probability that the weather is dry n days later.

a.) Show that Pn satisfies the recurrence relation Pn = (2p 1)Pn1 + (1 p) for n = 2, 3, 4, ....

b.) Solve this for Pn explicity in terms of p and n and explain why your answer makes sense for the special cases when (i) p = 0 (ii) p = 1/2 and (iii) p = 1.

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