Draw a Set card from the deck.

Problem

Draw a Set card from the deck.  Before looking at it, guess what it is.
What are the probabilities that you guessed correctly on exactly 0, 1,
2, 3, 4 of the four properties (shape, color, number, shading)?

-- Tom Magliery <mag@ncsa.uiuc.edu> (June 1995)

[ Editor: posted as "a Set puzzle (actually just another boring old
counting problem)" in the announcement of the Set Home Page.  The game
of Set uses 81 cards, one for each combination of three values of each
of the four properties; see competition/games/set. ]