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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.

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> 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.

Well, obviously if we have a better prior, then that's better. But assuming no other knowledge, and especially if we think that other people's priors could be intentionally misleading, this rule seems to offer the best estimate.

Generally speaking, you never truly have "no prior knowledge". Some relevant past experience, or "common sense", or something tips you away from "all probabilities are equally likely". I think this rule is rarely a best estimate.

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