Important evolutionary events such as the Cambrian Explosion have inspired many attempts at explanation: why do they happen when they do? What shapes them, and why do they eventually come to an end? However, much less attention has been paid to the idea of a “null hypothesis” – that certain features of such diversifications arise simply through their statistical structure. Looking back from our own perspective to the origins of large groups such as the arthropods, or even the animals themselves, will expose features that look causal but are in fact inevitable. I will review some of the features of such statistical structure with particular reference to the Cambrian Explosion and other famous radiations. A particularly interesting aspect of such bias is the effect on molecular clocks, and I will show why molecular clocks are likely to overestimate times of origins of large clades such as the animals and angiosperms.