RULES of the GAME: There are three inverted cups, one of which hides a valuable diamond. A contestant chooses one of the three cups at random (Move One). At this point, the probability of success, i.e., choosing the diamond, is 1/3.
Monty Hall, who knows where the diamond is, must eliminate one of the empty, unchosen cups, leaving only two cups on the table (Move Two).
If the contestant always switches cups (Move Three), then the chance of winning will double --- from the original 1/3, to 2/3. Using the graphic on the left, we review these points.
Move One: One of the three cups is chosen. The graphic shows all three possibilities. The first column shows the case when the diamond is chosen, and the last two columns show when an empty cup is chosen.)Move Two: Monty Hall eliminates an empty cup. Note that in the first column, Monty Hall must leave one unchosen empty cup, even though he has two choices of empty cups to eliminate. In the second two columns, Monty has only one choice of empty cup to eliminate and must leave the one unchosen cup with the diamond.
Move Three: Switch choice of cups to the other one remaining. If a switch is always made, then, as the graphic clearly shows, a winning choice of cup(column one) becomes a losing cup. Conversely, an initial losing choice (columns two, three) converts to a winning choice.