In Gin-Rummy, what are the chances to start with a Gin?
1 in ~118,000, according to 2+2 forum post.  Using a brute force, that checked for Gin each of the 15.8B hands.
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: