It'll be the same with automated driving software. All of the liability for accidents that occur in cars using the software will be concentrated with the software developer, instead of distributed amongst all of the drivers. That might be too much liability to make the business viable.
The good news for us drivers: our insurance rates should drop once the cars are fully automated and manual control isn't even possible. At that point we'll just be passengers, and we shouldn't need to pay insurance.
However the bigger point is I doubt you'll own a car. More likely there will be lots of driverless taxi companies, and it'll just be very convenient to use your phone to call a taxi to take you where you want. In this case, the insurance cost is folded into the cost of hire.
In urban areas, people travel light, but that's not the case in sub-urban and rural areas.
For bigger kids there are some sorta options like RideSafer vests, but with a baby you're just screwed.
Zipcar has the same issue; our use of it dropped hugely once kids appeared, because it's just too painful.
In other words, robots are held to a higher standard than humans.