Ye've lost yer marbles

It has been a long year at Puzzlania Academy, and the sweet smell of spring (and graduation) is in the air. Ah, how wonderful life is. Unfortunately for you, there's one little thing left before you can savour the aroma of graduation : the Puzzlania Ordinary Level Examinations. Rumour has it that your examiner this year is none other than Dr. M. Grant, the famous mathematician.

As you enter the dimly-lit examination room for the first time, you are unsurprised to see Dr. Grant sitting behind an old desk.

"I assume you are familiar with the format of your POLEs?" he asks. You nod. "There will be seven exams this year. You will have as much time as you'd like to write each one." He gestures toward a table with a pencil and piece of paper on it. "You may begin."

You sit down at the table and begin reading the paper. There is only one question written on it, and it seems rather simple.

I have 729 black marbles and 243 white marbles. I will draw one at random and discard it into a fiery pit. I will then keep drawing more random marbles and discarding them until one doesn't match the previous one. That marble will not be discarded, but put back in the bag. The marbles will be reshuffled and the process will start over.

Example:
I draw a black marble and discard it.
I draw another black marble and discard it.
I draw a white marble and shuffle it back into the bag.
I draw a black marble and discard it.
I draw a white marble and shuffle it back into the bag.
I draw a white marble and discard it.
I draw another white marble and discard it.
I draw another white marble and discard it.
I draw a black marble and shuffle it back into the bag.

After a little while, eventually only one marble is left. What is the probability of it being black?

"This is easy," you exclaim, "the answer is..."