Statistically, you have to ride for something like 90 years to be guaranteed a serious crash
Statistically, you’re never guaranteed one. But in any case a chance of less than 100% is still a chance. Let’s assume the probability approaches 1 in 90 years, then even in 30 years of riding to the shops you’ve got a 1:3 chance of a serious crash on the way to the shops. I’d wear a helmet for that!
Equivalent distance, not equivalent risk
Quite so, I was assuming constant risk for all cycling which isn’t true. In fact, I’d say urban cycling is more risky due to lots of cars containing drivers who are having to look out for lots of things; and the prevalance of such things as bollards and dozy pedestrians.
Conversely, if you are on an MTB ride there might only be a couple of difficult or fast downhills and the rest could be climbing at 4mph or cruising traffic-free. I don’t think helmet off-road and no helmet on-road is a sensible criterium because road is probably more dangerous than all but the fastest or most careless off-road. And “just going to the shops” is no better.