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?
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: