More data addressing some of the comments to Dan Piponi's empirical observations.

Gerry Myerson wanted a range of primes just up to a power of 10.  The 664,579th prime is 9,999,991.  I'm going to be dropping digits so I'll start with 101, the 26th prime.  The conjecture holds for the 26th to 664,579th primes with 324,133 digits 0 and 570,148 digits 1:

[![distribution of digits][1]][1]

Note that the 0 count is the same for all four histograms.  Dropping the first digit still puts 1 on top with 490,132 but now last place is the 323,065 digits 5:

[![distribution of digits except first][2]][2]

Dropping the last digit still has 1 on top with 404,049 and 0 last, but the nonzero digits are almost uniform:

[![distribution of digits except last][3]][3]

Finally, for Gerald Edgar's idea of dropping the first and last digits, i.e., looking at just internal digits, the results are almost uniform for all digits.  Here 0 is actually first (still 324,133) and 9 last with 322,604:

[![distribution of internal digits][4]][4]


  [1]: https://i.sstatic.net/MaXTirpB.png
  [2]: https://i.sstatic.net/C9R1u5rk.png
  [3]: https://i.sstatic.net/oTCPKisA.png
  [4]: https://i.sstatic.net/y8fmUi0w.png