Acting on an anonymous phone call, the police raid a house to arrest a suspected murderer. They don't know what he looks like but they know his name is John and that he is inside the house. The police bust in on a carpenter, a lorry driver, a mechanic and a fireman playing cards. Without hesitation or communication of any kind, they immediately arrest the fireman.

a. How do they know they've got their man?

Because he was the only man in the house playing cards. The other three players were all women.

Explanation:

The fireman being arrested without hesitation or communication of any kind inside the house by the police was through direct observation. The police did not know what he looked like but they knew that his name was John. The fact the his name was John indicates that the murderer was potentially a man.

When the police busted the house, they might have observed that the fireman was the only male playing cards with the carpenter, the lorry driver, and the mechanic. The possibility of arresting him directly might have been that he was the only male inside the house.