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Comment by Terr_ | Hacker News Reader

To try to put it into words:

We don't actually know the mix-rate of what's in the urn. It's possible we just pulled the only red ball and all the others are green... However the fact that our first n=1 sample happened to be red (and not something else) gives a small (and varying) amount of confidence towards red-heavier mixes rather than the red-scarce ones.

We can use that to figure out which mixes are slightly more-likely to be in there (based on our tiny amount of data after assuming they begin equally-likely) and then for each of those determine "if it's mix X, then what are the odds of our next draw", and combine the two layers of probability.

In this case, the bias towards presuming the mix is fundamentally reddish outweighs the presumption of "removing one red makes the greens the majority."

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Yes, that's exactly right. Paraphrasing what another poster wrote about this on Twitter: "You would rather go fishing in a lake where you just saw someone else catch a fish."

Also, the logic continues as you draw more balls from the urn. If the second ball is ALSO red, then you have even more evidence suggesting that N was selected in such a way as to make red the overwhelmingly more likely choice. Thus the chance of the third ball being red is even higher than the 66% probability in the second draw.

To extend the fishing metaphor, the premise of ~100 balls is also important: If each lake could only sustain one (good) fish, you would actively avoid anywhere someone had already caught one from.

I guess the counterpoint to this is as the number of balls goes from 100 to infinite, it becomes increasingly harder to predict, correct?

No, the higher the count goes the closer it becomes to the single removed ball not mattering at all, so the only thing that matters is the information you got about which urn you're selecting from.

According to the original poster on Twitter:

"One cool thing is that this probability doesn’t change if you replace 100 in the statement of the problem with some other number (as long as it’s at least 2), as these arguments show."

Source: https://x.com/littmath/status/1752167731682492522?s=46

That's "what's the probability you've caught the world's meanest fish".

> However the fact that our first n=1 sample happened to be red (and not something else) gives a small (and varying) amount of confidence towards red-heavier mixes rather than the red-scarce ones.

I wouldn't characterize this as a small amount of confidence, as conditional distribution of the mix-rate after the first sample drastically differs from the prior.

Originally each mix-rate has 1/101 probablity. After the sample having a mix with n reds in it has the probablity 2n/(100101).

I agree. This is a beautiful summary of the logic I used

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