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In Gin-Rummy, in the start of a round players draw 10 cards.

What is the probability to draw 10 cards such that all the cards are part of some meld?

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1 in 308,984, according to this analysis, which includes source code so you can verify.

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    Does that include being able to pick up a visible discard to make Gin on your first turn? I'm inclined to think it doesn't, because I've done that twice, and I only play gin for an hour, once a week, for the past 4-5 years. – Zeiss Ikon Jul 15 at 12:07
  • it does not. Just the first 10 cards. Getting a gin in the first turn (out of the first 11 cards) is a better (much harder question – Cohensius Jul 15 at 12:14
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1 in ~118,000, according to 2+2 forum post. Using a brute force algorithm, that check each hand out of the 15.8B hands if it a Gin hand.

1 in ~117,000, according to the book How to Win at Gin Rummy: Playing for Fun and Profit which state that there are 136,694 Gin hands out of 15,820,024,220 hands. See page 65: gin hand - starting amount of deadwood

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