Your airbag won't always go off. But that does not mean that it is not working. Many cars have sensors that only trigger them in case of a severe impact that the car's passive safety systems could not deal with.
Topgear, a BBC car show, did a test where they crashed a brand new minivan head-on into a 11-year old minivan (same model, much earlier model year).
Both minivans had airbags, and they were testing the effect on crash dummies in the minivans.
In the 11 year old minivan, the driver's airbag inflated during the impact.
In the brand new minivan, the driver's airbag did not inflate.
But in the 11-year old one , the car's safety cage was unable to handle the impact and the crash dummy suffered other injuries that would have been fatal
In the brand new minivan, not only did the airbag NOT inflate, but the crash dummy suffered almost no injuries.
They got someone from Renault, who made the minivans, explain why the airbag did not inflate in the new minivan. He explained that the sensors embedded in the minivan detected that the force of the collision was less than the maximum force the safety cage and sealtbelt system in the modern van were designed to bear, so it did not see a need to inflate the airbag.
This technology has been widespread for nearly 10 years.
What I'm getting at, is that in a low-speed crash, up to around 40-50km/h, a modern airbag is unlikely to inflate. It is designed to protect you in high-speed crashes.