If something has a 5% chance of success, and I’ve failed 19 consecutive times, the next time independently still has a 5% chance of success, fairly low, rather than a guaranteed 100% chance of success in view of the previous attempts. How does this relate to updating percentages in light of new evidence, or is that something separate?
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You’re mixing up two concepts. You’re starting with the assumption that the chance of success is 5% – that is, you *know* the probability going in. There’s nothing for you to update with new observations; you’re already certain it’s 5%. Since you seem to be assuming the trials are independent (even if you aren’t saying so), the probability remains 5% regardless of past trials (since that’s what independent means in the first place).
If, on the other hand, you were not certain about the underlying probability, and you fail 19 times, your estimate of the underlying probability *would* correctly shift downward (since you’d be more likely to observe 19 failures if the probability were low than if it were high). [Bayes’ theorem](https://en.wikipedia.org/wiki/Bayes%27_theorem) is the mathematical law that tells you how to update your beliefs in light of new evidence, at least in simple cases like this (it can be tricky to apply in real world settings). The underlying probability is still fixed, but your *estimate* of that probability is changing, and (under some relatively weak assumptions) will approach the true value as you gather more data.
If your question is “how does it adjust if it fails a bunch to ‘get back to 5%'” – it doesn’t. The streaks just tend to cancel out in the long run.
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