> Assuming that the normal rate of battery fires after collisions is one per million, the probability of observing such a fire after only 13 collisions is rather low.

What you're arguing is that the probability of it happening on or before the 13th time is less than it happening on or before the millionth time -- Which is self-evidently true of _any_ probability, and is a much weaker argument.

> perhaps the rate is not one per million

This is much closer to the argument you should be making. I.e. "Given that a purported one-in-a-million event occurred after only the 13th test, what is the probability that their estimate of one-in-a-million is accurate?"

(I don't know the math to answer that question, but I hope somebody that does will come along and chime in.)

Assuming I interpret the actual problem at hand correctly, this is how we formulate it statistically.

We have an event that occurs with probability p = 1/1000000, given an accident (of which there were 13).

The probability of having zero catastrophic events, given that we have observed 13 accidents, is (1-p)^13 = .99987

(For some real hair-raising fun, try this calculation with the success/failure rate of a typical condom.)

Note that this is a purely frequentist interpretation and ignores any Bayesian inference (ie, we assigning a weight of 0 to our Bayesian prior, which is atypical). It's not a very robust way of modeling the situation, for a number of reasons.

Given this XKCD article, I would suggest the probability of a battery fire after an accident is around 1 in 26.

Note that due to the lack of selection bias (battery fire was already among the top security concerns before the accident), we don't actually need a very stringent p-value, so we can be fairly confident that the probability is greater than 1/260.

But my point was that even with probability 1/260, it's likely to be much higher than for gas cars, or you wouldn't have to dig up 20-year-old articles about "freak accidents".

Gas cars have electrical fires all the time. We don't read about it because it's not news.

Admittedly, those cars tend to be poorly maintained, and not nearly new high end expensive vehicles.

> Statistically, there have been fewer fatalities with Model S than the average for the number of miles driven

Which of course it's not. But the general tone here of "move along nothing to see here" is transparently wishful thinking.

Of course a battery fire after hitting some road debris after only a dozen accidents is something to be concerned about. Maybe we don't know just how much yet but no one should be dismissing it offhandedly.