How could you modify the card game rating calculations here to take
LUCK into consideration.
http://www.silvercrk.com/rfaq.htm#calculate

The problem is that currently the system assumes everything is skill
based.  But the card games in question (Hearts,Spades and Euchre) all
involve an element of risk.  For instance an excellent player could
lose to a poor player just due to the cards they get.   So I'm looking
for some ways to take the luck of the draw  into account to make the
results more fair.

In any one particular hand in a game of, say, Hearts, luck will be a
factor.  However, in a game of Hearts up to a hundred points, there
will be at least several hands to be played.  The odds of several
hands in a single game of Hearts decided by luck would be so small
that the effect of luck would become negligible.

Hi silvercreek:

I'm not sure I see a realistic way that you can measure the "luck" of
a win/loss. And, I'm not sure why you would need to.

I play in several online card ladders and can tell you from experience
that luck doesn't have a great impact on overall ratings. A player
wins just as many "lucky" games as they lose over the long run and the
effect on their rating is nullified.

Now, *if* you still want to try to do this, I think you need to
realize that whether something is lucky or not is somewhat subjective.
Maybe you should leave it that way - how about a brief "questionaire"
after each game asking player whether they felt that luck was a big
factor in the game? If *all* agree (including the winner) then you
could modify the values of "o" and "w" in your rating equations
accordingly.

Other than that, I can't think of any way you can modify your system
that would be fair.

Hope this helps. 

websearcher-ga

Well there is and example of statisical the distibution of cards in a
4 player card game where the deck is split evenly:
http://www.durangobill.com/BrSuitStats.html

I wonder if getting an unlikely hand could be factored in a way that
would be less subjective.

Basically, it seems like what you want is a way to determine how
'good' a player's hand was.  This is likely to be fairly complicated,
and certainly depends on the game you're playing.  To get an answer, I
think you'll have to pick one game, and provide a precise set of rules
for that game.  Even then, it may be impossible to be entirely
objective about it.  For example, take hearts: it's obviously good to
get low cards, and also to be short-suited in one or more suits.  But
how important are these two factors relative to each other?  Also,
it's clearly bad to have, say, the king as your only spade.  I think a
formula that took all this into account would be very complicated. 
And it would get even more so when you consider that you can get a
good hand for shooting the moon.  In addition, the same hand could be
'good' or 'not so good' depending on how the other cards are
distributed among the other three players.  For example, if you plan
to shoot the moon, and you have seven clubs including the A and K, but
not the Q, then you can take all the clubs if each of the other
players has 2, but not if a player who has 3 clubs has the Q.  So
should your 'luck' calculation also take into account the hands of the
other players?
If you want a really 'fair' rating system that is influenced by luck
as little as possible, here is the best idea I can think of:  Find (or
write) a computer program that plays the game in question.  I think
this should be much easier for a game like hearts than for, say,
chess.  The program needs to play very well--not like the Windows
hearts game.  But again, since remembering the cards that have been
played, and calculating probabilities is easy for a computer program,
I bet there are programs that play hearts and can beat almost any
human player in the long run.  Presumably, the program works by
evaluating each of the possible moves (there are only a max of 13
possible on any one turn).  So what you could do is use the program to
evaluate each move that each player made, and use the results to
compute a rating.  The only problem with this is it would be pretty
much impossible to include any possible psychological aspect of the
game.  For example, a good human player might decide to shoot the moon
with with a hand which the computer program wouldn't, because the
human player could gamble on an opponent thinking "he can't be
shooting the moon, I've got 3 kings!".  Still, I think evaluating
plays made by humans to with a program that is successful at the game
is your best bet.