[286c8cb04bda]

If the chance of a catastrophic event happening is one-in-a-million, then it is equally likely to happen on the 13th and on the millionth coin toss.

> 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.)

[chimeracoder]

Time for some fun with binomial probabilities!

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.

[tedunangst]

I think the right test to apply is the Wilson test.