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. Show that the weather is dry on January 1, then Pn, the probability that it will be dry n days later, satisfies
Pn = (2p − 1)Pn−1 + (1 − p) n ≥ 1
P0 = 1
Prove that
Pn = 1/2 + 1/2 (2p − 1)n n ≥ 0