The Watchmen realise that no-one has ever said "It's practically a certainty, but it might just work!" and so attempt to increase the odds of Sergeant Colon shooting a dragon to exactly a million to one (eventually blindfolded, standing on one leg, his face covered in soot and singing 'the hedgehog song').
They fail. Fortunately the odds of surviving the dragon's counter-attack were exactly a million to one.
This one gets funnier in hindsight, as Vimes arrests Vetinari in Jingo.
Even more so when Sgt. Colon somehow manages to vocally command Carrot's central nervous system, and sent him marching away with a few army-styled orders. Especially when the narration notes that while Vetinari is speaking to Vimes, Colon is screaming at Carrot in the background to stop marching.
Carrot arrests the dragon, and then holds back the mob. Can't have prisoners killed by a mob.
When a few men are discussing the dragon hunt and what the reward should be, one of them suggests it should be to marry the princess, or at least the patrician's daughter. Another informs him that he has no daughter, just an aunt and a dog. After the rest of the men leave, one comes back and asks what kind of dog it is.
One of Carrot's first acts after joining the Watch is to arrest the Head of the Thieves' Guild for being a thief. Unfortunately for Carrot's ideals, the city now has organized crime - which makes the crime rate easier to control, since police have to work ten times as hard to cut down on crime even a little, while criminals just have to work a little less. "Next he'll be arresting the head of the Assassins' Guild for bloody well killing people!"
"It could have been an innocent bystander, sir." "What, in Ankh-Morpork?" "Yes, sir." "We should have grabbed him, then, just for the novelty value."