There were 21 7+ quakes in 2010 by my count (and admittedly I could be off one or two each way as I was editing for posting).
That's 84 days in a two-day window each way, or about a one in four chance (23 percent) that a given 188 day cycle will hit that window.
Make that over four cycles and you have one in sixteen.
I'm not too sure that you're not right about the odds. For the sake of argument, lets use your 21 7+ count per year for 4, 188 day cycles.
21/365 = 0.057534 7+ earthquakes a day for 752 days (188 x 4 cycles).
That works out to a total of 43.2 7+ earthquakes during the entire 752 day period (752 x 0.057534).
The odds of a 7+ occurring on any one particular day during that period are 1 in 17.4 (752 / 43.2). Close to what you came up with.
But, (what I think forgot to include) the odds of a 7+ occurring on exactly 188 day intervals during the 752 day period are more like 1 in 139.2 (1/(17.4 x n))?