user1485: The famous Shannon number, 10^120, an estimated lower bound on the number of possible chess games, is based on the assumption that a typical game lasts 40 moves.
In games between skilled human players, this average game length seems to be about right (see this question), but if we want to count all possible chess games, then we also have to include the vast majority of games where the moves don't have any purpose or sense whatsoever. Such a game with random moves would surely be likely to last more than 40 moves?
So what is the average game length of a random game? This should be easy enough to simulate on a computer.

user4362: I reran the script from the answer GloriaVictis linked.
For a twenty-five thousand games the average length was 357 ply. 

user9230: An interesting question with which I've done significant exploration.  My program Symbolic can generate random games using the /dev/urandom pseudorandom generator to any length.  For a recent run of ten million random game, the average length was 342.064 ply.
[] rg 10000000
   Checkmate  1527544 0.152754
  FiftyMoves  2241451 0.224145
Insufficient  5358614 0.535861
  Repetition   258221 0.0258221
   Stalemate   614170 0.061417
Average ply length: 342.064
Maximum ply length: 974


1485: <p>The famous <a href="https://en.wikipedia.org/wiki/Shannon_number" rel="nofollow noreferrer">Shannon number</a>, 10^120, an estimated lower bound on the number of possible chess games, is based on the assumption that a typical game lasts 40 moves.</p>

<p>In games between skilled human players, this average game length seems to be about right (see <a href="https://chess.stackexchange.com/questions/2506/what-is-the-average-length-of-a-game-of-chess">this question</a>), but if we want to count <em>all</em> possible chess games, then we also have to include the vast majority of games where the moves don't have any purpose or sense whatsoever. Such a game with random moves would surely be likely to last more than 40 moves?</p>

<p>So what is the average game length of a random game? This should be easy enough to simulate on a computer.</p>
 4362: <p>I reran <a href="https://math.stackexchange.com/questions/845424/the-expected-outcome-of-a-random-game-of-chess">the script</a> from the answer GloriaVictis linked.</p>

<p>For a twenty-five thousand games the average length was 357 ply. </p>
 9230: <p>An interesting question with which I've done significant exploration.  My program <em>Symbolic</em> can generate random games using the /dev/urandom pseudorandom generator to any length.  For a recent run of ten million random game, the average length was 342.064 ply.</p>

<pre><code>[] rg 10000000
   Checkmate  1527544 0.152754
  FiftyMoves  2241451 0.224145
Insufficient  5358614 0.535861
  Repetition   258221 0.0258221
   Stalemate   614170 0.061417
Average ply length: 342.064
Maximum ply length: 974
</code></pre>
