Expected number of years in the English Football League for a newly promoted team

Question

(For the sake of this question, let us assume that the Premier League is still the top division of the EFL). The English Football League is a professional football (soccer) league consisting of four divisions with promotion and relegation between each division. 20 teams play in the top division (The Premier League, in this case), while 24 teams play in each of the other three divisions (The Championship, League One, and League Two).

Promotion means that a team plays in the division above the following season, and relegation means that a team plays in the division below the following season. Every season, the following occurs:

• 3 teams are promoted from the Championship to the Premier League
• 3 teams are relegated from the Premier League to the Championship
• 3 teams are promoted from League One to the Championship
• 3 teams are relegated from the Championship to League One
• 4 teams are promoted from League Two to the League One
• 4 teams are relegated from League One to League 2
• 2 teams are relegated out of League Two to non-league status
• 2 teams are promoted from non-league football to League Two.

Say you are one of the two teams fortunate enough to be promoted to the Football League for next season; how many seasons on average can you expect to stay in the EFL before being relegated out of it again?

Answer 1

Let's introduce $\mu_P,\mu_C,\mu_1,\mu_2$ denoting the expected seasons a club will stay in the EFL if it starts in Premier League, Championship, League One, League Two respectively.

Then we find the following relations:

• $24\mu_2=24+4\mu_1+18\mu_2$
• $24\mu_1=24+3\mu_C+4\mu_2+17\mu_1$
• $24\mu_C=24+3\mu_P+3\mu_1+18\mu_C$
• $20\mu_P=20+3\mu_C+17\mu_P$

To be found is actually $\mu_2$.