Ok, I did use the word symmetry in there

:D

Note that more generally you should ask for p(new information | scenario) and apply Bayes Rule, but anthropic-style information is a special case where the value of this is always either 0 or 1

But it's not entirely special, which is interesting. For example, say it's 8:00 and you have two buckets and there's one ball in one of the buckets. You have a 1/2 chance of getting the ball if you pick a bucket. Then, at exactly 8:05, you add another bucket and mix up the ball. Now you have a 1/3 chance of getting the ball if you pick a bucket.

But what does Bayes' rule say? Well, P(get the ball | you add a third bucket) = P(get the ball) * P(you add a third bucket | get the ball) / P(you add a third bucket). Since you always add a third bucket whether you get the ball or not, it seems the update is just 1/1=1, so adding a third bucket doesn't change anything. I would claim that this apparent failure of Bayes' rule (failure of interpreting it, more likely) is analogous to the apparent failure of Bayes' rule in the sleeping beauty problem. But I'm not sure why either happens, or how you'd go about fixing the problem.