(For more information on the probable cause concept, see How much "probable cause" do cops need? For an explanation of the standard needed for a detention, rather than an arrest, see What is reasonable suspicion?)
To establish probable cause, police officers must be able to point to objective circumstances leading them to believe that a suspect committed a crime. A police officer can't establish probable cause by saying only something like, "I just had a hunch that the defendant was a burglar."
Judges, not police officers, have the last word on whether probable cause exists. A police officer may be sincere in believing that the facts establish probable cause. But if a judge examines that same information and disagrees, then probable cause does not exist (or did not exist, if the question is being decided after an arrest).
Note that probable cause may have existed at the time of an arrest even if the defendant didn't actually do anything wrong. Put differently, an arrest is valid as long as it is based on probable cause, even if the arrested person is innocent.
The million-dollar question is: How much information do police officers need to convince a judge to issue an arrest warrant or to justify a warrantless arrest? In general, probable cause requires more than a mere suspicion that a suspect committed a crime, but not as much information as would be required to prove the suspect guilty beyond a reasonable doubt.
Because probable cause is an abstract concept, a firm definition of it is evasive. Courts have to determine case by case whether there is or was probable cause for an arrest.
Example: Officer Furman arrives at Simpson's Jewelry store moments after it's been robbed He sees broken glass inside the store. A man claiming to be Simpson, the owner, is on the scene. He holds what look like keys to the store and seems distressed. He tells Furman that a man, approximately 6'5" tall and weighing over 300 pounds, held up the store at gunpoint and escaped with rings and watches in a small brown paper bag. A few minutes later, less than a mile away from the jewelry store, Officer Furman pulls a car over for speeding. The driver matches the description of the robber, and on the seat next to him is a small brown paper bag and a couple of watches with the price tags attached. Though Officer Furman did not see the robbery itself, the driver matches the unusual physical description of the robber and has property that looks like what Simpson said was missing. Furman has probable cause to arrest the driver.
Example: Same case. Assume that the person claiming to be Simpson, the jewelry store owner, was actually the robber's accomplice. The accomplice gave Officer Furman a phony description and then fled after the officer drove off. The driver pulled over by the officer for speeding is later able to prove that he is the lawful owner of the watches that the officer saw on the seat. In this scenario, the broken glass corroborated "Simpson's" statement that a robbery had occurred, and Officer Furman had no reason to doubt the word of the person claiming to be Simpson. Thus, the officer had probable cause to make the arrest, even though the information turned out to be incorrect.
Example: Officer Seesit pulls over a car and its three occupants for speeding. The officer searches the car (with the driver's consent) and finds baggies of cocaine stashed behind an armrest in the back seat. All three occupants say they didn't know the cocaine was in the car. Seesit has probable cause to arrest the car's occupants. In the absence of evidence demonstrating that the cocaine belonged to a specific occupant, the officer could reasonably conclude that all of them knew about and possessed the cocaine. (Maryland v. Pringle, 540 U.S. 366 (2003).)