Hurkyl, I admire your persistence in try to explain this.
The statement presented by the author quoted in the OP is unambiguous and clearly correct, but the brief manner of presenting an incomplete statement in the second observation clearly has caused a lot of confusion.
I believe the following statement is a way to say what the original author intended to say but which avoids the confusion and likely would not have even lead to this thread having been started in the first place.
"If you roll n dice the likelihood of your getting any SPECIFIC sequence is identical to the likelihood of your getting any OTHER specific sequence, but if I don't see you roll the dice and you tell me you got a pretty random jumble of numbers, I'm going to find that easy to believe because it's quite likely that's what will happen, but if you tell me you got something like all ones, I'm going to find that pretty odd because it's unlikely. On the other hand, if I gave you a random sequence of numbers and asked you to roll that SPECIFIC sequence, I would find your getting it to be exactly as unlikely as your getting all ones."