One million players participate in a game that has 10 levels. At first, all players are at level 1.
At the end of each turn, each player rolls a twelve-sided die, numbered 1 to 12. Player advance a level if he gets a higher number of its current level number. Otherwise, it remains at the same level.
Warning! At 10, "advance a level" means return to level 1!
After some time, the number of players in each level tends to stabilize.
What will be the average number of players at level 10? Be rounded off to the nearest integer.
Any help to that problem would be appreciated. it's not a homework problem.
You need to set up a Markov Chain and find the transition probability matrix, based on the rules described. Then, find the stationary distribution. Then, average number at level 10 (or any other level) is found from this stationary distribution.
I'm not giving the details because they are widely available.