Well, 300,000 miles, one non-fatal accident (with, again, a human at the wheel - but we'll ignore that for now).
Now, I pulled these numbers of a set of google searches. There was a fairly wide range of stats, so I took a bit of an average:
Insurance industry assumes one accident CLAIM per 17.9 years (lots of minor accidents go unclaimed, but we'll ignore them too). Average of 15,000 miles per year per driver. Thus, an average of one accident per 268,500 miles per human driver.
Of course, while the human driver stats are numerous (and this is why insurance is expensive!) the self driving car stats are not. Only one accident with new, unrefined technology in 300,000 miles... and that with a human in control of the car.
That said, your example? That's where a self driving car is much, much better than a human. A human driver with a pregnant woman giving birth, woken up in the middle of the night is going to be tired, highly agitated and distracted and definitely not at his best. The self driving car isn't tired. It doesn't care what time it is. The self driving car will be aware of the speeding racers - and know their exact speed, trajectory, and likely path - sooner than the human driver will, as these are very simple computations to make. The self driving car is indifferent to the passenger; which is also important. It's not distracted, worried, or anxious.
Of course, there certainly are cases where that's just not good enough, extreme emergency cases. That's why all these self driving cars can be driven in manual mode. You've always got that option if need be.
Obviously, routes being set aside for autonomous vehicles will be safer, but routes mixed will be safer than pure-human routes, because autonomous cars are simply safer than human driven cars overall.
I've been rearended while stopped at traffic lights six times in the last twenty years; every time due to an inattentive driver. None of those would happen with an autonomous car.
Finally, yes, mechanical/electronic failure can result in crashes. Just like it can with human drivers - sticking accelerators, for example, failing steering linkages, brakes, etc. Software problems? No different than a human driver having a heart attack, stroke, seizure, getting stung by a bee, etc - those all happen all the time. There's no real difference there.