A more thorough explanation:

The hot pursuit doctrine is a legal principle that allows police officers to enter a private residence without a warrant when they are in pursuit of a suspect who has fled from a public place. This means that if a suspect runs into their home, the police can follow them inside to make an arrest.

However, this exception to the Fourth Amendment's search warrant requirement is only applicable if the police have probable cause to make an arrest and if the arrest was already in motion. In other words, the police cannot simply enter a home without a warrant just because they suspect someone of committing a crime.

Hot pursuit is typically used in situations where the police are chasing a suspect who has just committed a felony and is trying to escape. If the police believe that the suspect poses a threat to public safety or is likely to destroy evidence, they can enter the suspect's home without a warrant to prevent these things from happening.

Imagine that a police officer is chasing a suspect who has just robbed a convenience store. The suspect runs into a nearby apartment building and locks himself in his apartment. The police officer can use the hot pursuit doctrine to enter the suspect's apartment without a warrant to make an arrest.

Another example would be if a suspect is being chased by the police and runs into their own home. If the police believe that the suspect is armed and dangerous, they can enter the home without a warrant to prevent the suspect from harming anyone.

These examples illustrate how the hot pursuit doctrine can be used to allow police officers to enter a private residence without a warrant in certain situations. However, it is important to note that this exception to the Fourth Amendment's search warrant requirement is only applicable in limited circumstances and is subject to review by the courts on a case-by-case basis.