> There were 9 crewed missions and 1 almost killed its crew (I will omit Apollo 1 for now). I could argue that Apollo had a 1 in 20 chance of killing a crew.
That's not how risk analysis works.
Let's say every Apollo mission had gone flawlessly, and no one even came close to dying. Would you then say that the risk of death for future missions would be zero? No, of course not.
I thought I'd look this up. If you've had 9 successful attempts, assuming nothing has changed between them and no other prior knowledge about success probability, then Laplace’s Rule of Succession says the probability of the next mission being a success is about 83.3%, i.e. there is a 1 in 6 chance of failing next time.
> and no other prior knowledge about success probability
This phrase is misleading, as Laplace's Rule of Succession is equivalent to assuming a uniform Bayesian prior over all values of p. That is, before any experiments, a 50% chance of success. Depending on the situation, this may be roughly accurate or wildly wrong. You cannot appeal to this rule to resolve the situation.
