> Statistically, it could be said you're at a greater risk than someone who got in
> an accident yesterday.
Statistically, no, since each event (trip) is independent. The odds of getting into an accident on any given trip are the same whether it's the first one or the 100,000th one (inexperience and old age notwithstanding (withstanding, you get a bathtub curve)). It's like flipping a coin: it's very unlikely that 157 coins spun consecutively will come down heads 157 consecutive times (unless your name is Rosencrantz), but the odds for each individual coin toss are still 50/50.
Also, driving isn't a pure game of random chance. Driver skill, state of mind, chemical level, etc. all come into play, all of which have a profound effect on the odds.
Besides, it's not as if it's some kind of freakish event to have driven so long without getting in an accident. Lots of people have done it.
And actually, try telling that to the insurance companies, who penalize you heavily for an accident (my rates doubled, and they essentially told me in no words to get stuffed when I moved). They believe that an accident is an indicator of future accidents (which it is, but only after a certain point).