20Sided Dice

The "house" rolls two 20sided dice (each numbered 120) and the
"player" rolls an identical 20sided die. If the player rolls
a number on his die between the two numbers the house rolled,
then the player wins. Otherwise, the house wins (including
ties). What is the probability of the player winning?

The game is changed slightly: two of the 20sided dice are
rolled, and the player must choose one of them for his number.
Then the third die is rolled to determine the house's second
number. What should the player's method be for making his
choice, and what is the probability of winning now?

If the player is given an option of either choosing a house
die or rolling his own, what is his best strategy, and what
is the probability of winning?
It may be easier to find results for smaller dice first.
Source: rec.puzzles, with original extensions.
Solution
Mail to Ken