Look at it from another perspective, namely your own.
You are "ball number 7". When you are pulled out of the barrel, what is the chance that you will be number 7? Why, 100% of course. What is the chance you will be taken out of the barrel? Well, that chance will be much higher when your barrel has only 7 or 8 balls then when it has 100 billion, the Carter scenario says (paraphrasing of course). So probably you are in a small barrel (i.e. extinction will occur soon).
This is where the scenario has gone wrong, in my view. We don't know and have no reason to assume that only one ball is taken out of the barrel. If all balls are taken out, ours has to be taken out as well, and the fact that ball number 7 has been taken says nothing at all about the size of the barrel.

Let's look at it again, but from a completely different angle.
I am filling a barrel with numbered balls slowly and in order. One ball drops every second. After ten seconds, you pick a ball. It's number 7. What does this say about the total number of balls that will end up in the barrel?
The Carter scenario says that 95% certain, this will be only 20 or so. I guess it is plain for everyone to see that we have no information whatsoever to make such a statement. The only thing we know is that a) there will be at least 7 balls, as I have picked ball seven, and b) there will even be at least 10 balls, as so much seconds had passed (the observer can of course see the balls dropping). Nothing more.

Relating this to the original presentation of the scenario: you have two of those slowly filling barrels. One will stop at 100, the other at 10000. After ten seconds, you pull out ball 7. Does this help in any way to know which one will stop at 100 and which at 10000? No.

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Knowledge is a curse, but ignorance is worse