The risk factor is calculated _per mission_ from what I understand. You can have three accidents in a row and nothing for decades but the risk itself can still be 1 in 30.
- Historical. Low N as you say. (Even though each mission and spacecraft is different and they're spread out over time, there's value in this)
- Bureaucrat number; absurdly low, but looks good to politicians etc
- Engineering estimateNo, but it means that to ensure that I do better on my next set of coin tosses I need to beat 3 in 4, not 1 in 2.
That assumes a fair coin. The fact is you don't know what the odds were of getting heads or tails for that particular coin, all you know is that you got 3/4 heads. And in this analogy, a few hundred coins have every been made, in maybe a dozen styles, none of which have been fair, so you have no good reason to believe that this particular coin should have 50/50 odds of landing heads up.
We are not dealing with the tautologically true statement that we are assuming the 1/1000 estimate is correct and thus the odds are 1/1000 no matter what we measure. We are dealing with whether or not we can safely reject the hypothesis that the true odds are 1/1000 based on the actual observation of 1/12.
Billions of coins have been minted, and flipped a countless number of times, and we can do the physical analysis of coins such that we know the odds of a coin not being fair, without deliberate intervention to make them such, are astronomically low. As such no one is going to reject the hypothesis that a coin is fair based off of a small number of coin tosses. Hell even if you got 10 heads in a row, while the odds of that sequence is 1 in 1024, we would probably conclude it was luck rather than that the coin was flawed.
For spaceships on the other hand, those priors don't exist. We need to look at just the data from this particular test. The odds of a 1/1000 event occurring in the first 12 attempts is 1 in 84. For rejecting the hypothesis that a mass produced coin is fair, those odds aren't bad; but for rejecting the null hypothesis that the apollo capsules were just unlucky it's way over the reasonable threshold.
The public doesn't care if Apollo had a theoretical risk rate lower or higher than 1/12, what they saw was that 1/12 missions resulted in the death of the crew. The NASA administrator explaining that their estimated risk was only 1/1000 doesn't change the real-world perception or outcome.
FWIW, the 1/12 is also actually off, the long-term mortality rate for Apollo astronauts is high.
But so is the 1/1000, Nasa's own estimates were so bad that they decided it was bad optics to keep doing them - https://ntrs.nasa.gov/api/citations/20190002249/downloads/20...
But then, Apollo 1 was after all the first mission on the Saturn V. I think we should assess even its pre-launch risk much higher than the rest of them. Similarly Artemis II has a much higher risk than the subsequent ones will have.
Of course Apollo would have likely had a better average if it had continued, but the risk of the Apollo program, as executed, included things like the first flight of the Saturn V.
If the final empirical mortality result of the Artemis program is 1/30 or less, it will be better than Apollo in that statistic.
A comparison of acceptable mortality is where this discussion began. If Apollo was acceptable at 1/12 (We did it, it was apparently acceptable as the program was not cancelled due to mortality rate) then an acceptable mortality of 1/30 is stronger than Apollo, not weaker.