Police don't actually need "probable cause" to pull you over for California DUI. But they do need "reasonable suspicion"--which in practice is hard to distinguish from "probable cause."
Probable cause is a key concept in American constitutional law. Under the Fourth Amendment to the U.S. Constitution, law enforcement officers need probable cause to arrest a suspect for a crime--including California DUI. Probable cause can be roughly defined as a reasonable belief that a crime has been committed, based on objective facts.
But in the landmark case of Terry v. Ohio (1968), the United States Supreme Court held that something less than probable cause is needed for a traffic stop--including a traffic stop for suspected DUI. The Terry standard is usually described as "reasonable suspicion."
But "reasonable suspicion" is really not that dissimilar from "probable cause." It also needs to be based on objective facts that would lead a reasonable person to suspect that a crime was taking place. Like "probable cause," it can't just be a hunch or a "bad feeling." An officer can't pull you over for DUI because he or she had a "bad feeling" about your car.
One way of characterizing the difference between probable cause and reasonable suspicion is that probable cause is a reasonable belief that a crime has taken place, whereas reasonable suspicion is just what it sounds like: a suspicion that a crime may be taking place.
Police also need reasonable suspicion to conduct a DUI investigation (e.g., to administer field sobriety tests)--but the standard there is somewhat higher than what's required to pull you over for DUI. In other words, there must be more concrete signs that you might be under the influence for the officer to proceed with an investigation.
And officers do still need probable cause to arrest you for DUI.
It's also important to note that the probable cause/reasonable suspicion does not apply to DUI sobriety checkpoints. Officers may pull you over at these checkpoints even without any reason to believe you may be DUI, as long as the checkpoints are conducted in accordance with certain legal requirements.