# Markov Chain - "Expected Time"

The Megasoft company gives each of its employees the title of programmer (P) or project manager (M). In any given year 70 % of programmers remain in that position 20 % are promoted to project manager and 10 % are fired (state X). 95 % of project managers remain in that position while 5 % are fired. How long on the average does a programmer work before they are fired?

Solved it through the transition matrix where the equations are, m(P) = 1+ 0.7 m(P) + 0.2 m(M) m(M)= 1 + 0.95 m(M) So m(M)= 20, and m(P)= 16.67,

I hope someone can confirm this. Thank you

• Perhaps you should explain and justify in detail (not winging it) how you determined these equations, and how their solution provides the solution to the question asked. Mar 17, 2018 at 21:12
• You should also try another method, such as evaluation of the probability of being fired in n steps for n from 1 up to a large enough number, by computing the n step transition matrix. Then use these values to compute the expected number of steps (years) until firing. Mar 18, 2018 at 1:33

• Programmers stop with probability $p=0.3$ after each year, so stay programmers for $1/p = 10/3$ years on average
• Project managers are fired with probability $p=0.05$, so say in their job for $1/0.05 = 20$ years on average.
Thus, the average length of a programmer's career seems to be $10/3 + 2/3 * 20 = 16.67$. So I'm siding with you on this one.