# Probability of a disaster

Can you help me with this Markov chain problem? This should be simple, but my brain is not working well.

Let $\{{x_t}\}_{t=0}^\infty$ follows a Markov chain. Each $x_t$ can take two values: {G,B} (good or bad. bad represents a disaster). The transition probabilities are two unknowns:

Pr(G | G) = x

Pr(G | B) = y

You know the following:

-$x_0 = G$

-Expected duration of a disaster is d periods.

-The average frequency of disasters is f disasters per 100 periods

What are x and y?

Thank you.

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## 1 Answer

The duration of a bad period is geometrically distributed with mean 1/y=d. Likewise, the mean duration of a good period is 1/x hence a cycle good period+bad period has mean length c=1/x+1/y. In the mean there are 100/c=f such cycles in a period of length 100. Solving these equations yields x and y as functions of d and f. The fact that x0=G is irrelevant.

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Thank you Didier! –  Toan Nov 9 '11 at 16:48