This is a variant of the argument here. I am not sure which version I like more. This version is lacking one distracting feature of the original, but I kind of like the original more.
You're suddenly informed by an angel that a countably infinite number of people have just each rolled a fair and independent indeterministic die. In the case of each of these people, you should surely assign probability 1/6 that that person rolled six. The angel then adds that infinitely many people rolled six and infinitely many didn't. This doesn't surprise you—after all, that was just what you expected from the Law of Large Numbers.
The angel then adds that he divided up all the die rollers into pairs, one member of each pair having rolled six and the other not. This doesn't seem to be a big deal. After all, you knew that they could be thus divided up as soon as you heard that infinitely many rolled six and infinitely many didn't. The mere fact that they were so divided seems irrelevant.
Finally, the angel transports you to meet each pair of paired die rollers, pair by pair. In each pair, you now know for certain that one person got six and the other didn't. Let's say that at some point you meet Jennifer and Patricia. What probabilities do you assign to each having rolled six? You can't stick to your old assignment of 1/6 to each roller. For if you did that, you'd be violating the probability calculus, since you know for certain that exactly one of the two got a six, so you better have P(Jennifer got six)+P(Patricia got six)=1. But you can't go for any asymmetric assignment either, since your evidence regarding Jennifer and Patricia is exactly symmetric. That leaves you with two options: you must refuse to assign any probability to the probability of Jennifer getting six and to the probability of Patricia getting six, or you must assign probability 1/2 in both cases. And since this reasoning applies to everybody, you'll be assigning 1/2 to everybody or no-probability to everybody.
Assigning 1/2 to everybody is untenable, because you could then re-run the scenario with a different way of partitioning (say, into groups of three, two of whom rolled six, and one who didn't).
So it seems that you just need to refuse to assign probabilities. But from which point on? As soon as you learned that there were infinitely many people rolling independent fair dice, you already knew that some partitioning like the above was possible. Intuitively, you didn't learn anything important at any subsequent step. (You did genuinely learn something when you learned there were infinitely many sixes and infinitely many non-sixes, but since that information had probability 1 before you learned it, it shouldn't have affected your probabilities significantly.)
One way out is to deny the possibility of engaging in probabilistic reasoning in a scenario where there are infinitely many independent copies of an experiment. This way out insists that from the beginning, as soon as we knew about the infinitary aspects of the experiment, we should have refused to assign probabilities. (But I worry: What if the experiments are sequential? Shouldn't it be possible to engage in probabilistic reasoning then? And yet one can make a similar story work in sequential contexts.)
Maybe, though, one can switch from assignments of 1/6 to a no-probability at the last step. When I meet Patricia and Jennifer, I meet a highly biased sample of a pair of die rollers: after all, in an unbiased sample, I would have no guarantee that exactly one of the two rolled six. The bias takes a form that cannot be probabilistically handled by me: I have no probability distribution on ways of pairing individuals that assigns a probability to Patricia and Jennifer being paired together. And where there is bias that cannot be probabilistically handled, I must suspend judgment. This is not a very attractive way out. But maybe it's still the right one? (Notice, though, that if you are told ahead of time that you will be meeting people in pairs like this, then you will violate a plausible generalization of van Fraassen's reflection principle if you initially assign 1/6 and then switch to no-probability.)