What rules of poker are played on the Enterprise?
It's not Texas hold em.
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In "All Good Things" Picard deals and says:
"So. Five card stud, nothing wild. The sky's the limit."
In "The Measure of a Man" they also play poker. Riker says:
"The game is five card stud, nothing's wild."
But the next time they play in this episode Pulaski says:
"The game is seven card high/low with a buy on the last card. And just to make it interesting the man with the axe takes all."
Looks like the dealer can call whichever play he wants.
They play 5 card stud.
More recently, in the television series Star Trek: The Next Generation, the crew is sometimes seen playing five-card stud, often an officer's game hosted by Commander William Riker and usually entered by Data (as part of his efforts to become more human), Worf, Geordi LaForge, Dr Beverley Crusher, Deanna Troi and occasionally by other crewmembers. An entire set of scenes in the seventh season episode "Lower Decks" cuts between the 'senior officers' poker game and the 'junior officers' poker game played by characters not seen in other episodes, before resulting in the junior officers being cleaned out by a civilian known only as 'Ben', who then joins the senior officers game. The game is played most notably during the final scene of the series finale "All Good Things", when Captain Jean-Luc Picard for the first time joins the others as the dealer and, after revealing that he was quite the card player in his youth and being told he had been always welcome, gives the final words of that episode and of the series overall: "Five-card stud, nothing wild... and the sky's the Limit" as the camera pans overhead the table.
Edit: As pointed out in the comments, the dealer seems to be able to call whatever game type they would like (wild cards etc.) but the game-type played seems to always be a variant of 5-card stud.
We see both stud and draw poker played.
There are a variety of poker games in the cluster of Stud poker. In stud, betting begins before the player has a full hand. Variations on the stud game include 5-8 cards, which ones are face up, and how many are dealt before the first round of bids, and what's wild.
There's also Draw Poker; we see some hands of draw poker. Variations are fewer - how many cards, how many betting rounds, and what's wild.
Note that Texas Holdem is a variant of stud, with the face ups being shared.
Wildcard variations are extremely common in casual draw. High, Low, and High-low are victory definitions; high is the default; low means worst hand takes the pot; High-low means the highest and lowest hands split the pot. "Takes All" and "Busts" define certain cards that override the normal scoring; Take all means that the card holder wins the pot, and busts means that no matter how good the hand, they lose if they have that card in hand. These same variations are less common, but still frequent, in Stud.
Draw is easier on the props people; the hand is present, and they only have to stack for the draws.
In at least one game, it's called stud, but played as draw... the betting begins only after the players all have a full hand.
Also, draw is easier on props guys if a specific hand is required - the deal is done face down, so they just swap for the needed hands during a cut.