> This means that every state is recurrent, i.e. will happen if you wait long enough.
It doesn't mean that the expected frequency of every state in the chain is equal. Moreover, the frequencies are very different.
The interesting part of the experiment is not that each state is reachable. The state where someone has all the money - 1 is reachable. The state where all have the same amount of money again is reachable. But if you leave these people alone in a room for some time and open the door unexpectedly, you will almost sure find out that they have very different amount of money.
I ran an numerical experiment and I get that the average distribution of money is for 5 persons with 5 coins each is:
11.04 6.38 4.03 2.44 1.11
i.e. if you enter the room at random, the one with more money will have on average 11.04 coins, not almost 24 coins
If you count the number of times they have
11±1 6±1 4±1 3±1 1±1
coins, you get a 21% frequency.
If you count the times someone has 24 coins, you get only a 0.00125% frequency. If you count how many times someone has at least 90% of the coins you get 0.0131% frequency (10x more).
This is difficult to compare, because each classification has a different amount of states, but nevertheless, it's clear that an almost linear distribution is more common than a very concentrated distribution.
For 10 persons with 10 coins I get that the average is
28.77 19.11 14.25 10.98 8.53 6.57 4.93 3.53 2.27 1.06