If your cat finds a mouse inside the house and you find a hole in a closet where it enters, will you become infested?

This is Not my house! And I do not have a cat. But I was wondering whether all homes in which the owner(s) has found a dead mouse in their cat’s mouth and how it entered via a closet hole that leads to the outdoors will soon be infested with mice? Or do many people just live with a handful of mice year after year and it never gets bad? I am talking about people who refuse to hire an exterminator.