The Rules of Fizzbin

(Fizzbin is a fictional playing-card game created by Captain James T. Kirk in the Star Trek episode #49 "A Piece of the Action." The game was "invented" while Kirk, Spock, and Dr. McCoy were being held hostage on Sigma Iotia II. Kirk spontaneously created a confusing card game to distract the henchmen guarding them. The rules are intentionally very complex.)

The Rules of Fizzbin
Each player gets six cards, except for the player on the dealer's right, who gets seven.
The second card is turned up, except on Tuesdays.
Two jacks are a "half-fizzbin."
If you have a half-fizzbin:
a third jack is a "shralk" and results in disqualification;
one wants a king and a deuce, except at night, when one wants a queen and a four;
if a king had been dealt, the player would get another card, except when it's dark, in which case he'd have to give it back.
The top hand is a "royal fizzbin". The odds of getting one are "astronomical." (When Kirk asked Spock what the odds are, Spock truthfully replied that he had never computed them.)
memory-alpha.org: Fizzbin
wikipedia.org: Fizzbin

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Fans of the old Star Trek will remember that Fizzbin is a poker-like game supposedly played on Beta Antares IV. You are dealt a 6-card hand from a standard deck. A "Fizzbin" is a hand with two pair plus two unmatched cards. (For example, king of spades, king of hearts, 9 of hearts, 9 of diamonds, 5 of clubs and ace of hearts). If the pairs are both 10 or higher (that is, 10, jack, queen, king or ace), then it's called a "Royal Fizzbin". [OK, OK, I'm making most of this up -- but so was Captain Kirk.]
www.ma.utexas.edu: Problem 3. Fizzbin - What is the probability of being dealt a Fizzbin (either royal or regular)?