What is the intuition behind Laplace’s Law of Succession?

Suppose you periodically try to read Quora, but the site is sometimes down or unresponsive. In your first 20 tries, the site has been down 5 times. What is the probability that the site will be down the next time you try? Laplace's Law of Succession says that the answer  is 6/22, not 5/20.
In Rule of succession on Wikipedia, you can read an explanation of this that states the mathematical assumptions, uses Bayes' rule to infer a distribution for the probability for Quora to be down, then integrates over that distribution to find the probability that it will be down on your next visit. However, in a recent public Facebook discussion (I am puzzled by Laplace's Law of… – Brienne Yudkowsky), several people came to a much simpler demonstration.
As a model, suppose that when you make an account, Quora gives you a random number between zero and 1 uniformly, called your reference number. Then each time you try to log on, they roll a new random number, also between zero and one uniformly. If they roll higher than your reference number, they serve you the site completely operational. If they roll less than your reference number, they don't serve the site, or comments fail to load, or links to question pages are unresponsive, etc. Quora is down.
Overall, there have been 21 rolls so far (1 for your reference number, 20 for your first twenty  visits). When you roll again, you could roll lower than all the first 21 rolls, lower than all but 1 of them, lower than all but 2 of them, etc. There are 22 possibilities. All these possibilities are equally likely because there are now 22 total rolls, and if we put them all in order, by symmetry the last roll has an equal chance at any spot. Six of these possibilities correspond to Quora being down (final roll is lower than all but anywhere from 0 to 5 of the first 21 rolls), so overall, the probability that Quora will be down next time you try to log on is 6/22.
Disclaimer: all websites and statistics about their operation appearing in this answer are purely fictitious. Any resemblance to real websites and their operation, living or dead, is purely coincidental.

Answer by Mark Eichenlaub:

Suppose you periodically try to read Quora, but the site is sometimes down or unresponsive. In your first 20 tries, the site has been down 5 times. What is the probability that the site will be down the next time you try? Laplace's Law of Succession says that the answer  is 6/22, not 5/20.
In Rule of succession on Wikipedia, you can read an explanation of this that states the mathematical assumptions, uses Bayes' rule to infer a distribution for the probability for Quora to be down, then integrates over that distribution to find the probability that it will be down on your next visit. However, in a recent public Facebook discussion (I am puzzled by Laplace's Law of… – Brienne Yudkowsky), several people came to a much simpler demonstration.
As a model, suppose that when you make an account, Quora gives you a random number between zero and 1 uniformly, called your reference number. Then each time you try to log on, they roll a new random number, also between zero and one uniformly. If they roll higher than your reference number, they serve you the site completely operational. If they roll less than your reference number, they don't serve the site, or comments fail to load, or links to question pages are unresponsive, etc. Quora is down.
Overall, there have been 21 rolls so far (1 for your reference number, 20 for your first twenty  visits). When you roll again, you could roll lower than all the first 21 rolls, lower than all but 1 of them, lower than all but 2 of them, etc. There are 22 possibilities. All these possibilities are equally likely because there are now 22 total rolls, and if we put them all in order, by symmetry the last roll has an equal chance at any spot. Six of these possibilities correspond to Quora being down (final roll is lower than all but anywhere from 0 to 5 of the first 21 rolls), so overall, the probability that Quora will be down next time you try to log on is 6/22.
Disclaimer: all websites and statistics about their operation appearing in this answer are purely fictitious. Any resemblance to real websites and their operation, living or dead, is purely coincidental.

What is the intuition behind Laplace's Law of Succession?

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