I've thought about this problem a long time, and the only thing that allowed me to wrap my head around it was to draw it out on a small piece of paper. Draw th

I've thought about this problem a long time, and the only thing that allowed me to wrap my head around it was to draw it out on a small piece of paper. Draw three rows of three rectangles each. Now draw a star (representing the prize) in row one, box one, another star in row two box two, and finally a star in row three box three. This depicts all possible "game states". Now we will play the game three times, but in each case you will always choose the first box in each row as your first chosen door. Now the host will open one of the other doors. In the first row, you have two choices to open, so pick one and draw a circle in it. This door is open revealing nothing. In the second and third rows, you must not open door with the prize, so you choose the empty box in each row. Now go through and tally up how many times you would have won if you had not switched doors and stayed with your original guess (1/3). The tally how many times you would have won had you switched (2/3).