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Comment by gkoberger | Hacker News Reader

When trying to get popular opinion for, say, an election -- sure.

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.

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Forgetting about all of the (completely legitimate) concerns with this comparison aside from statistics, we can get a general idea from simple statistics how convincing having the first crash at 100 million miles should be.

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).

Doesn't stopping the clock after 1 fire just penalize them, not benefit them?

That said, I also think the 1 fire is the problem here. Just think about how that relation changes with 2 fires.

The chance goes from 0.000000010 to 0.000000020

If I had the cash I would still purchase a Tesla after the second fire as well.

Thanks for converting that to "fires per mile".

More miles would have been driven, so less. I'd be happy after the hundredth fire. They'll be even safer by then.

It probably penalizes them, assuming the no-fires-for-100m-miles was not a fluke. There's no way to know for sure without knowing the true distribution of fires per mile.

By your "hole in one" argument, Tesla is "doubtful" to keep up their pace of one fire per 100 million miles. In other words, their "true" fire rate is actually less than one per 100 million.

If you assume that fires are normally distributed, it's equally likely to be a larger or smaller length of time to the next one.

I think a Poisson distribution is what you're looking for here. Roughly, if events happen independently and with a fixed probability per time interval, you get a Poisson distribution. Poisson distributions apply to a lot of things, so it's a very useful distribution to know about.

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.

Presumably it will go up to some degree over time as the average age of tesla's fleet converges with the industry average, as older cars are more likely to experience malfunctions/breakdowns for obvious reasons.

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_...

You don't know that and this is the whole point of the "hole in one" argument.

You example just reinforces the point.

Please don't argue against statistics (mathematical information based on fact) when you don't understand them.

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?

Why do you believe a larger sample is needed?

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`.

If I get the math right (assuming a Poisson distribution) then a week ago, based on 0 fires, they could've said with 90% confidence "The chance of your Tesla catching on fire is less than 2.3 cases per 100 million miles". Not 0; but not very high as well.

> 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.

This wasn't after driving 100 miles. Aren't you off by 6 orders of magnitude?

This is related to something called the Doomsday Argument, and it was recently the topic of an xkcd what-if blog:

http://what-if.xkcd.com/65/

By all accounts it comes down to the old bayesian/frequentist battle lines.

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