It has been estimated that a young healthy cat could easily kill a thousand mice in a year. Most homeowners will be happy to know that their own houses are unlikely to have a thousand mice in a year, or in ten years.
So in short, if you do own a cat, you probably won’t have mice around, or not for long. Rodents are not stupid, and they will tend to avoid a house where a cat lives. Unlike the cartoons, where the wily mice always get the better of the cat, in real life rodents either get eaten or move on to a catless home.