20-Sided Dice

  1. The "house" rolls two 20-sided dice (each numbered 1-20) and the "player" rolls an identical 20-sided 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?
  2. The game is changed slightly: two of the 20-sided 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?
  3. 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.

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