Marion,
If the house move is random then;

1) Player selects a cup. (sticks with choice, 1/3 chance to win)
2) House turns a random cup from the remaining two, (1/3 chance to win)
3) Player switches to remaining cup.
(but only has 2/3 chance of getting to this point in the game.)

If case three arises, and the player always switches, his chance to win is 2/3, but only in 2/3 of all the games played. (The house wins 1/3 of the games when it selects at (2).)
Thus by always switching the player wins 4/9 (2/3*2/3) of all games played and the house wins 5/9 (1/3 + (2/3*1/3)) OR (1 - 4/9).
These are marginally better odds than the 1/3 : 2/3 the player gets if he sticks, but nowhere near the 2/3 : 1/3 if the house always reveals an empty cup. ie The house now has the edge, as it wins more that 50% of the games

I hope that;s right ;)

Wolfgang,
B"

I hope that's right too ;)

AndyG