Applications of the Ergodic Theorem for Markov Chains
I need help to solve these 2 exercises:
1) A professor has N umbrellas. He walks to the office in the morning and walks home in the evening. If it is raining he likes to carry an umbrella and if it's fine he does not. Suppose that it rains on each journey with probability p, indipendently of past weather. What is the long run proportion of journeys on which the professor gets wet?
2) An opera singer is due to perform a long series of concerts. Having a fine artistic temperament, she is liable to pull out each night with probability 1/2. Once this has appened she will not sing again until the promoter convinces her of his high regard. This he does by sending flowers every day until she returns. Flowers costing x thousand pounds, 0<=x<=1 bring about a reconciliation with probability sqrt(x). The promoter stands to make 750£ from each successful concert. How much should she spend on flowers?
These are taken from Markov Chains by J.R. Norris and should be solved using the ergodic theorem and the strong law of large numbers. I can't solve them :(
Thanks in advance!!!