Outside of Israel, we start saying "v'tain tal oomatar livrachah" ("and provide dew and rain for a blessing") in the amida on the evening of December 4th or 5th (depending on the year). Chanukah starts the evening of the 25th of Kislaiv. How often does Chanukah start before we switch to "v'tain tal oomatar livrachah"?
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I have a book here that lists the Hebrew-calendar dates for starting "v'sen tal umatar" for the years 5750 through 5851. Counting, I see that in 26 of those 102 years (25%) Chanuka starts before that date. (And in two of the years they start the same night: 5787 and 5833.)
There can't be a constant ratio for this, as the date of starting v'sein tal umatter changes (on the english calendar) every 3 out of 4 centuries (thus if you look in the Tur you will find a different date than the one in the shulchan aruch, which are both different from what we do). This is because we follow two different opinions in Maseches Rosh Hashana for how to calculate the exact length of a year, and v'sein tal umattar follows the "less exact" calculation. Theoretically, in a few thousand years v'sein tal will start very late and never be before chanuka.