Continuity mistake: In the scene showing Groucho being chased across the cars as they enter the race course, in the shot from behind you see him running along the rail only to slip, the following shot is from the front and shows the same slip he's already made.
Continuity mistake: In front of the betting windows, Chico sells Groucho the code books and master code books, then tells him he needs the entire set of Breeder's Guide books. Groucho pulls out his money and said "Give me all four", but Chico hands him five books.
Other mistake: Chico sells Groucho a paper with the name of a horse who is sure to win. It says "ZVBXRPL." Chico explains that the name is in code and then sells Groucho a code book. Groucho looks up the name and says "ZVBXRPL, page 34" but he is looking at a right-hand page. Page 34 would be a left-hand page.