Other mistake: The reporting of the horse race used to sting Doyle Lonnegan stops as soon as the fake FBI agents raid the bookmakers. J J Singleton, who is reading the results of the race, stops when he sees the raid taking place. However, that was supposed to be a live broadcast from the racetrack itself. It should have continued. Lonnegan isn't stupid - he would pick up on something like that immediately.
Plot hole: Doyle attends Gondorff's betting shop three times, and he listens to the announcer calling three races from three different race tracks - Narragansett in Rhode Island, Belmont in New York, and Riverside Park in Missouri. He cannot possibly miss the fact that the same announcer calls all three races! J.J. Singleton, the race caller, has an instantly recognisable voice, and Doyle wouldn't be fooled for a second. Each race track would have had its own announcer.
Suggested correction: There are a lot of things Doyle is processing when he's in the shop. There's a lot happening, and he has a lot on his mind, and increasing pressure and stress each time. It's quite possible that he wouldn't notice the accent of the announcer, something he has no reason to doubt.
Rubbish. During his first two visits, he sits quietly listening to the race announcements. On his second visit, he would recognize J.J. Singleton's distinctive voice and would realize something was very wrong.
Other mistake: Why is the cashier sitting behind bars in the mock-up bookies when he just steps out of the side to go and fetch Gondorf? If anyone wanted to get at the cashiers they could just walk around the side where there is no door.
Suggested correction: It's partly psychological. Also, anyone going around the bars would be slowed down, more than simply reaching over, giving the cashier the opportunity to react.
Character mistake: At Duke Boudreau's bar, the potential recruits for the sting are told, "Gondorff is setting up a con on the North side". Later in the film, Agent Polk tells Lt. Snyder, "Gondorff is running a con on the South side". The actual address for the betting hall is not given, but it is just down the alley from Klein's, which does have a South side address.
Suggested correction: It's possible recruits were given the wrong information on purpose until they'd been cleared for participation. Misinformation is a good way to confuse and delay any unwanted attention.
This does not correct the posting. "Polk" tells Snyder that Gondorff is running a con on the NORTH side, which is incorrect. He has absolutely no reason to try to mislead Snyder.
Suggested correction: This isn't a mistake. The person "operating the radio" could easily have switched it off because of the commotion. Lonnegan would simply suppose this is what had happened. He is not going to instantly suspect someone is faking the whole thing.