Firing on the very first compression is fairly rare, and certainly should not be considered as typical.
The more common situation is that the initial kick (or starter engagement) starts the crank shaft turning. At some point, sooner or later, at the top of a compression stroke, the engine fires. If enough energy is imparted to the system (chemical burn energy plus remaining kinetic energy from the initial kick), then the engine starts and you are off to the races.
How many times the engine has to turn before it fires is different in just about every situation, depending on mixture temperature, engine oil temperature (affects oil viscosity), and a host of other variable.
Many years ago, I worked on standby diesel power systems for nuclear power plants. These are intended to supply emergency power to the core cooling pumps if primary power is lost. It is absolutely essential that these standby units start when required. The machines my employer built were powered by diesel locomotive engines. They were equipped with six starters on the flywheel, working in pairs. Four of these were compressed air starters, working in pairs. When the signal to start was received, the first pair would attempt to start the engine. If after about 2 seconds the engine had not fired, the first starter pair were disabled and the second pair attempted to start. If that failed also, the first pair were energized again to attempt a start, and if that failed, then the second pair would try again. If all that failed to start the engine, the third pair were energized. These were different. The had an explosive charge serving as a high pressure gas generator, and thus were a one-shot only, last ditch resort. The expectation was that, within 15 seconds after the initial signal to start, the engine would be started, the speed stabilized at running speed (900 rpm), the field applied to the generator, the speed stabilized again, and the load motors on the pumps would be started. That was a very demanding spec to meet!
