How long are extra-inning games?

From the Baseball Reference Stat of the Day: the number of Major League Baseball games which went exactly n innings this year are as follows:

n	10	11	12	13	14	15	16	17
number of n-inning games	97	58	34	16	9	3	1	2

(The table in the original post actually gives twice the number of games which have gone at least n innings; I've done the obvious manipulations to get this table.)
If you look at this sequence, one thing that sticks out is that it's roughly geometric. Each number is just about half the number before. What this indicates is that a game has roughly the same probability of ending in any extra inning, namely about one-half. The geometric distribution is "memoryless"; this indicates that at the beginning of the nth inning, for any n ≥ 10, the probability of the game going one more inning is 1/2, two more innings 1/4, three more innings 1/8, and so on. The expected number of additional innings to be played is always the same -- around two.

This isn't all that surprising -- every extra inning "looks the same" when it starts. It wouldn't surprise me to see that things start to change, say, around the fifteenth or sixteenth inning when teams just run out of pitchers -- it wouldn't surprise me to learn that scoring rates go up in, say, the seventeenth inning because whoever is on the mound either is the guy you never let play or the guy who usually pitches an inning or two that's in his fifth inning of work -- but there's not enough data from just a single season to say anything for sure.

But if we look a bit closer at the data, it does look like something like that happens. To be somewhat more precise, there were 220 extra-inning games played this year, and 465 extra innings; thus at the beginning of the tenth, one should expect a game to go 2.11 more innings. There were 123 11-inning-or-more games played this year, and 245 innings numbered 11 or higher; thus at the beginning of the eleventh a game should go 1.99 more innings. Similar numbers for the beginning of the twelfth, ..., seventeenth are 1.88, 1.84, 1.73, 1.83, 1.67, 1.00. The distribution isn't quite geometric, and the expected number of innings remaining in an extra-inning game actually does decrease as time goes on.

(Also, in general the fifth and sixth innings are surprisingly high-scoring. I blame the fact that starters seem to fall apart a lot in the fifth or sixth these days. I'm not saying I could do better, but these guys make a hundred times what I do, so they should be held to higher standards.)

I actually saw this in action on July 25th, when the Phillies played the Nationals and won in fourteen innings, of which I only saw the first nine. My father and I went to the game, and I believe my father had never left a baseball game early in his life. But we left after the ninth, because we had to pick my mother up at the train station. She said that she wouldn't have minded waiting, say, an inning or so. But you never know how long the game is going to last, which is why we left... and if we had stayed around for the rest of the game she would have been waiting at the train station for two hours. You always think the game will end soon. And it usually will. But sometimes it doesn't.