The data on 100 million is probably enough to compare with the 3 trillion miles. The populations don't need to be equal to compare them, just big enough that they are random and distributed enough.
However, there was one fire over 100 million miles. The problem isn't the 100 million, it's the 1 fire. This wasn't a controlled experiment, either -- they just stopped the clock as soon as the first fire happened, and multiplied. A week ago, they could have said "You have exactly a 0% chance of your Tesla catching on fire" and have been right by this logic.
To think of it another way -- let's say you get lucky and get a hole-in-one your 10th time golfing. Does that mean you'll have 10 hole-in-ones if you golf 100 times? Doubtful.
EDIT: Also, don't forget that Elon is mixing numbers. There's no fire if someone doesn't run over something. All these numbers show is that the average driver is more likely to run over something. Of course Tesla drivers run over fewer things -- there are no 16 year old kids texting while driving a Tesla... yet.
That said, I also think the 1 fire is the problem here. Just think about how that relation changes with 2 fires.
If I had the cash I would still purchase a Tesla after the second fire as well.
Please don't argue against statistics (mathematical information based on fact) when you don't understand them.
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.
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.
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?
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`.
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.
The median car in the US is ~11.6 years old[1], while Tesla's oldest vehicles were released in 2008, and the vast majority of their fleet was sold in the last couple years.
Obviously, the massive differences between electric powered cars and internal combustions engines means that they may never reach parity, but unless Musk has figured out a way to beat entropy, its pretty safe to assume that older cars will break down/suffer leaks/explode more than newer ones.
[1]https://www.polk.com/company/news/polk_finds_average_age_of_...
But since the number of cars is increasing, it's not a Poisson distribution; if the chance per car per time is constant, you'd expect the time to the next fire to be shorter.
For the purpose of our simple modeling, suppose that there is a constant risk per mile of the car catching fire, making an exponential model reasonable. Under this model, observing the first fire at 100 million miles would give a 95% confidence bound on the rate of fires of about one fire every 33 million miles.
If we're comfortable with the stated rate of about one fire every 20 million miles for other cars, then this would give a 95% confidence upper bound on the Tesla's rate of fires at about 60% of a normal car's rate. This isn't the 20% that Elon's statement would imply, but it does suggest a difference (which could just be due to other problems with the comparison).
This wasn't after driving 100 miles. Aren't you off by 6 orders of magnitude?
By all accounts it comes down to the old bayesian/frequentist battle lines.
Of course, I'm exaggerating in the other direction. What we really should be calculating is the odds that Teslas burst into flames less often than the average car, given that the average car does so every 20 million miles and the first such event in a Tesla was at the 100 million-mile mark. We're still failing to account for the fact that the average Tesla is newer and probably better-kept than the average car, but it would at least be a reasonable start.
I don't know enough statistics to perform this calculation, but I would like to see how it is done.
For an exaggerated example, if Tesla had driven a billion miles and had 0 fires, you shouldn't say that there's not enough data - you definitely would have enough data to say that the chance of fire is below the gas-car rate of 5 fires per 100 million miles.
I'm honestly curious how one models this type of thing statistically, and I am not convinced enough of its obviousness to just accept numbers that someone throws around.
Which is not true here, as the GP correctly notes.
[1] Assuming that you're not measuring average age or measuring 'who is at home', but if you want to see, say, the average political opinion of total USA population, which tends to correlate with age.
Furthermore, how do you gather the data needed to correct the polling numbers without being able to accurately poll people in the first place? Seems like a complete chicken-and-egg problem.
Of course, the refrain I often hear is that as long as you pick the RIGHT 1,000 Americans, it's as good as polling all 319 million.
If it's a proper random sample, then it's far better than sampling all 319 million because it's 95% accurate with about a 3% margin of error and vastly cheaper and actually practical; you can't poll 319 million people.
Polling 10,000 Americans would be vast overkill, in any case.
You can sample less than 2000 people and get 99% accuracy with a 3% margin of error for a population of 325 million. Increasing the sample size to 10k simply reduces the margin of error to 1.29%, hardly worth the extra sampling of 8k people.