Did you consider the possibility that it is in fact you who don't understand?
The Law of Large Numbers states that as more miles are traveled, the fires per mile will approach the expected value. It is entirely possible to have 10 fires in the next week.
We won't know what the expected fires/mile is until a much larger sample is collected. It will take years to prove out.
Why do you believe a larger sample is needed? At what sample size would you believe that the Tesla averages less fires than other cars?
Here's a question (for anyone in this thread arguing statistics) that has an actual numerical answer: given the information in the article, what is the probability that Tesla's indeed experience less fires per mile than other cars? If someone doesn't know how to calculate the answer to that question, he shouldn't be arguing here.
Let's see if I remember any of this. If we assume the null hypothesis that Teslas have 1 fire per 20 million miles same as other cars, then P(0 fires in 100 million miles) = 0.67% and P(1 fire in 100 million miles) = 3.4% from Poisson distribution. So the odds that you'd have no more than 1 fire in 100 million miles is 4%. So I reject the null hypothesis with a p value of 0.04. (edit: fixed values)
This seems a bit dodgy since I'm "designing the experiment" after the fact, but I'm not sure how to correct for that. Any Bayesian experts?
You need an exhauseted state space. You cannot empirically infer a legitimate probabliliy, eg n/100m miles) with only a single failure observation, if there are 100 possible ways to fail. At best you have data on (1) of (N) ways to fail, but surely in the case of car accidents N=large.
A total of 2,650 cars were delivered to retail customers in North America during 2012, 4,900 during the first quarter of 2013, and 5,150 during the second quarter of 2013
Assuming 13000 cars on the road, each car would have logged 9k miles to get 110m road miles, as quoted by Tesla. But we know from past industry experience, that road fires are proportionate also with fleet age.
So, if anything we the probability of a road fire is likely to go up as more failure modes are discovered (including by chance), and as the vehicles cycle through a normal working life.
Assume a Tesla vehicle has a constant risk of catching fire per mile. That is, we have a exponential distribution `P(catch fire after t miles | hazard rate) = P(t|a) = a exp(-at)` where `a` is the hazard rate (average fires per mile). Furthermore we'll assume an exponential prior on `a`: `P(a) = w exp(-wa)`. `w` is a parameter that expresses how much prior knowledge we have of `a`. In the limit `w=0` we know nothing at all, except that it's nonnegative.
Our data is the fact that we went 100 million miles before a fire, after which exactly one fire happened, so we want to find the distribution `P(a|t = 100 million)` which tell us everything we want to know about `a`.
Then use Bayes' theorem: `P(a|t) = P(t|a) P(a) / P(t) = aw exp(-a(t+w)) / P(t)`. The normalization factor `P(t)` involves an integral over `P(t|a) P(a) da` from 0 to ∞, which wolfram alpha tells me evaluates as w / (t+w)^2.
So our posterior probability is `P(a|t) = a (w+t)^2 exp(-a(t+w))`, but we can take the limit `w -> 0` at this point for a fully uninformative prior: `P(a|t) = a t^2 exp(-at)`.
So we can just set `t=100e6 miles`, and now calculate things like the expectation of the distribution: `E[a] = 2/t = 2e-8 per mile`. Or the probability that the hazard rate is less than other cars, which is the integral from 0 to 1/(20 million miles): `P(a < b) = 1 - exp(-bt) (bt + 1) = 0.96`.
Please don't argue against statistics (mathematical information based on fact) when you don't understand them.