> After 5000 runs, they were all 8.4-9.7%

This sample size is really small. I ran 100 million simulations in Nim[0] (takes around a minute). And distribution converges toward 9.09% on all positions equally:

    Average turns: 65.99609065001634
    Final position distribution:
     4: 9.095%
    11: 9.093%
     7: 9.091%
     3: 9.091%
    10: 9.090%
     9: 9.090%
     1: 9.090%
     8: 9.090%
     2: 9.090%
     6: 9.090%
     5: 9.089%
     0: 0.000%