Having a properly drafted and executed will does not prevent probate. Any assets you own at death in your name alone with no beneficiary must go through probate. Probate is the process that must occur in order to legally appoint someone to have access to and control over those assets after you die. If, at your death, you have a bank account in your name alone that holds $30k, only your officially appointed Personal Representative will be able to access that account (or even obtain any information about it). (Personal Representative is the “new-ish” term for executor.)
I often hear from folks who bring their deceased parent’s will into the bank in order to access his or her checking account. They are always confused when they are turned away. “I thought the whole point of having a will was to let me have access! I’m named in the Will as the Personal Rep, why wouldn’t they give me access?”
A will is, at its most basic, a set of instructions for the probate court. With an original, signed will, the court can officially appoint a Personal Representative, by issuing “Letters of Authority,” or “Letters Testamentary” (official paperwork allowing the appointment). With that document (not the will itself), the Personal Rep will be able to access all probate assets in order to pay expenses and make the distributions you have set forth in your will.
Having a will certainly makes the probate process easier than not having a will. But its existence is not the determining factor as to whether probate must occur. How you own your assets at death, and whether you have named a beneficiary, determines whether probate must occur.