Daniel has just started a tenure track faculty position as an assistant professor. Each year he has a 10% chance of choosing a different career path (i.e., quitting academia), a 20% chance of being promoted, and a 70% chance of remaining in his current position. Once promoted, he will never leave academia. The year he is promoted, he will start a service or research-based appointment with equal probabilities. He rotates between service- and research-based appointments each year. If he is working in a service-based appointment this year, then next year he will remain in a service-based appointment with probability 0.8 or move to a research-based appointment with probability 0.2. If he is working in a research based appointment this year, then next year he will remain in a research-based appointment with probability 0.4 or he rotates to a service-based appointment with probability 0.6. Assume that once he is promoted, he never leaves academia, he will never be demoted. Assume that if he leaves academia, he never returns to academia.

a) In the long run, what is the probability that Daniel ends up getting promoted?

b) How many years should Daniel expect to be in his current position (assistant professor)?

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