But looking at it from an insurers perspective, the notion that you do not get into an accident for 10 years in a row, when your probability of having one is 50% each year, means that, while the probability of not having one in the 11th year is still 50/50, not having one for 11 years in a row becomes less likely, thus, they may adjust the premium up to account for mean reversion.
However, their model already accounts for the fact that your discrete probability is pooled with other discrete probabilities and they need to invest the premium to give a return that will not only take in enough to pay out, but also do so with a profit over time. Thus, they really shouldn't care because the risk is pooled. In all likelihood, the reason why the premium goes up is because they think you are a captured good and can, and will, charge more until you get fed up and move.
First, applying a coin flip model to automotive insurance is invalid - coin flips are such that each one is independent of all others, whether a car accident occurs to a driver in a given year is highly dependent on the driver, and thus future years will have a strong dependence on past years.
Second, if we are talking about independent events like coin flips, then changing the wager (or insurance premium) based on the past history of coin flips is irrational. The coin will have the same probability of heads or tails on the current flip, as it did for anyone of the previous flips, hence the rational response is to hold the same wager. A concrete example:
You throw a fair coin (p(Heads) = p(Tails) = 1/2) 10 times - the first 9 flips turn up heads, the probability of that p(HHHHHHHH)=1/512. From there, p(HHHHHHHHHH) = 1/1024, but at the same time p(HHHHHHHHHT)=1/1024 - you still have an equally likely set of outcomes. Further any specific string of 10 heads and tails with have the same probability so p(HTHHHTTHHT) = 1/1024 as well.
As for the question of seeing 10 heads (or 10 tails) in a row and that being rare, it's because
1) Humans assign more interest in things that occur repeatedly, sensible given that we developed in an environment where most events have a dependence on the past.
2) This leads to humans not paying close attention to patterns that are not repetitive, so HTHHHTTHHT is not really any more interesting that THTTHHTHHT or HHTHHTHTT, so to our minds those outcomes can be lumped together in a set of boring things, while a small number of patterns such as all heads, all tails, every other heads, or 5 tails then 5 heads end up in a set of interesting things. Then it turns out that p(boring things) > p(interesting things), and likely even >>.
On the question of insurance: maybe.
Overall the expectation would be that drivers without accidents in past years are more likely not to suffer an accident in future years - this is the idea of good driver discounts.
However, as people age they slow down, and may tend to have more accidents. Also, a very long period of no incidents while driving may lead to complacency and lack of alertness.
Insurance companies base their premiums on empirically driven probability models that take a number of variables about the driver, the car, the area being driven, etc. into account to determine risk of claims. It's possible a driver with an 11 year long clean record moved into an older age bracket where risk of accidents is considered to be higher, or similar.