Actually, cats can keep all rodents out of the house. I have the laziest, most insolent feline that ever drew breath, but she does keep the mice population down to a bare minimum. Also, their extraordinary hearing will allow you to pick up on intruders - if you watch their ears prick up at the right time.
When you catch them doing something they know they shouldn’t be doing, the cat will just give you an evil look and wait to see if you will actually come over and make them stop, the dog will immediately stop and the look really guilty.
I cannot tell you how true that is! I now only have 2 cats. My 2nd cat Mary (I call her autistic Mary because there is something just wrong with her. I describe her the way most mothers do when they have an autistic child, but just can't pinpoint their finger on it (nor can the doctors until the child is older). However, ANY LITTLE noise, distractiom, inside ot out, she's on in 2 seconds.
When my place was under construction I had a mice problem. I had NO idea what a huntress she was. When there aren't mice it's moths, bugs, anything that moves and does not belong there.
My 2nd one Mimi is just a big fat fatty who lays around and comes to me when he wants affection. He just watches Mary the huntress do her thing.