How do small percentages continue to work out in the long run?

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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.

>How does this relate to updating percentages in light of new evidence, or is that something separate?

That’s something completely separate. New evidence would completely invalidate what you previously believed and whatever model you were using. All of that math needs to be thrown out the window because you have been wrong the entire time.

This has nothing to do with probabilities as you described, correctly, before that.

The key to this is whether or not each each attempt is independent of other attempts, or if it’s influenced by them. For example If you flip a coin 20 times in a row, each flip is independent of the previous or successive flips. They don’t impact it at all.

But if you are drawing a colored ball out of a bin, and removing that ball from the sample, that does influence subsequent picks.

It’s related to independent and dependent events.

A coin toss is an independent event. Every time I flip a coin, the odds are alway 50% for any given result. It doesn’t matter if the coin has landed a certain way previously; prior results don’t affect subsequent ones because the way a coin lands does not fundamentally alter the coin.

Things like pulling a certain sock color from a drawer are dependent. If I have 19 white socks and one black sock in a drawer, there’s a 5% chance that I pull the black sock on the first try. If I don’t put the first sock I pull back in the drawer for my second choice, now the odds of getting a black sock are up to 1/19 instead of 1/20. Those odds will keep going up as I remove more socks because I’m changing the situation. As there are fewer and fewer socks to pick, it becomes more likely that I get the black one. My future results depend on the changes to the system caused by previous results.