understanding the answer to your question requires a firm handle on statistics, and that's why your valid question has a complex answer.
the correct approach to analyzing this type of situation is to use the binomial distribution. here's how it works:
the binomial distribution has only two outcomes, the textbooks often call them success and failure. in my model, a car that has a collision fire in a year of operation is a "success" and a car that doesn't have a collision fire in a year of operation is a "failure".
to get binomial probabilities, you need three pieces of information:
1. p, which is the probability of seeing a successful outcome
2. n, which is the number of observations you are looking at, and
3. x which is the number of "success" outcomes encountered in the n observations.
first let's look at the "p"- the probability of seeing a collision related fire for one year of a typical car's operation. there are extensive statistics on millions of cars, hundreds of thousands of collisions, and tens of thousands of fire in the aggregate nfpa data set. so for me to say that there are 0.0000392 collision related fires per car-year (as i estimated in the kickoff post), i think it's very hard to argue that figure is very far off. there's simply too much data over too many years pointing to that estimate - the estimate is based on almost 130 million cars on the road! from a statistical standpoint what happens is that the estimated error is pretty close to sqrt(p*(1-p)/k) where k is the number of total vehicles for which i am estimating the p. the net result is that the .0000392 should be +/-10% of the actual answer for probability of a collision related fire in a year for an average automobile.
next, consider the "n". i know this is pretty darn accurate as well, as we know how many teslas were delivered and when, and we can pretty easily calculate "car-years" on the road. that's my 13,300. from the standpoint of a binomial distribution, 13,300 is a pretty large "n".
finally consider the "x". that's the 3 fires we observed in a tesla. we can be pretty darn sure that's accurate too (that is there's definitely not less than 3 fires because we have pictures and video of the car burning).
so what i am saying here is that we have a very good handle on the 3 inputs into the binomial distribution, the p, the n, and the x.
under these conditions, i can tell you pretty much *** exactly *** how likely it is that i will see 0 fires, 1 fire, 2 fires, 3 fires, etc.
just put in the following formula into an excel spreadsheet:
=binomdist(<<insert number of observed fires>>,13300,0.0000392,FALSE)
if you do this, you'll get these results (for various numbers of observed fires):
| observed fires | probability |
| 0 | 0.5937 |
| 1 | 0.3095 |
| 2 | 0.0807 |
| 3 | 0.0140 |
| 4 | 0.0018 |
| 5 | 0.0002 |
the answer is 59.37%
how about exactly one fire by now?
the answer is 30.95%
how about one or fewer fires by now?
the answer is 0.5937+0.3095 = 90.33%
and the important question: what is the probability i would see less than 3 fires?
that's the sum of the 0+1+2 fire values: 0.5937+0.3095+0.0807 = 0.9839%
that is, if tesla model s were as likely to have a collision fire as an ice automobile, there's a 98.39% chance that we would have seen 2 or less fires by now. this is virtually a mathematical fact.
at this point, i am realizing i have an error in my original post which i will soon fix - i came up with a probability that was a bit too high as i used 3 or fewer fires (i should have used 2 or fewer). i will go back and fix it.
that's why this third fire was so important, when there were 2 fires you still had an 8% chance of seeing that many fires. but now with 3, the probability is dropping of sharply that this is just a random variation we're seeing (of course that's still possible, it's just that the odds are now under 2%).
i'm really not sure how else i can express this view more clearly. if it's still confusing perhaps someone else who knows can chime in.
the sample size is not 3 - it is 13,300. the number of observed outcomes is 3. assuming my model inputs are correct or very close, the binomial distibution should correctly account for all of the facts properly in calculating probabilities.
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i haven't seen such a detailed level of data in my research, so i did the best i could with what i had =)
however, you could say that no matter what the separation is, it would almost certainly be worse for tesla. that's because collisions presumably include multi-car crashes + hitting road debris. so the number of fires related to hitting road debris is surely less than the number i gave, and that would make model s look even worse. the probability of a road-debris collision fire would be even lower, and then the odds of us seeing 3 model s fires being a random fluke would be even closer to zero.
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