When a cartridge fails to fire you should keep the gun pointed in a safe direction and wait 10 seconds before carefully unloading the firearm?

I am going to assume from the nature of the question that you are asking why this is often held as a wise procedure.

There are three basic types of malfunction that can occur with ammunition.

The first is a "Squib Fire" where the powder does not completely burn, and the bullet fails to leave the barrel.

The second is a misfire where the round fails to activate, and one hears only a simple "click"; indicating that the round is a dud.

The third type of malfunction is called a "Hang Fire" and is the reason for this procedure. When the first occur "Hang Fires" behave like misfires, the trigger is pulled and nothing happens. Then...a few seconds later...bang! A Hang Fire is the result when a primer fails to immediately ignite the powder, resulting in the round discharging some time after the trigger is pulled rather than within one second. Thus whenever you have a misfire you keep the weapon pointed in a safe direction and wait to be sure it is in fact a miss-fire and not a hang-fire. I have always advised my students to wait a full minute (assuming they are not in combat or what have you, or course.) And then to remove the dud round.