Days until a storm arrives in a weather chain

Question

Weather follows a daily chain. From Sunny, tomorrow is Sunny with probability 1/2 and Rainy with probability 1/2. From Rainy, tomorrow is Sunny with probability 1/4, Rainy with probability 1/4, and Storm with probability 1/2. Storm ends the chain. Today is Sunny. On average, how many days until the storm?

Accepted Answer

Let s and r be the expected days to reach Storm from Sunny and from Rainy; Storm has value 0. One day passes per step: s = 1 + 	frac12 s + 	frac12 r, \qquad r = 1 + 	frac14 s + 	frac14 r + 	frac12 \cdot 0. From the Sunny equation, 	frac12 s = 1 + 	frac12 r, so s = 2 + r. From the Rainy equation, 	frac34 r = 1 + 	frac14 s, i.e. 3r = 4 + s. Substitute s = 2 + r: 3r = 4 + 2 + r = 6 + r \;\Longrightarrow\; 2r = 6 \;\Longrightarrow\; r = 3, \qquad s = 2 + 3 = 5. Takeaway. Self-loops (a state that can stay put) just appear as a p\,x term on both sides; solving gives 5 days from Sunny to th…

Days until a storm arrives in a weather chain

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