Exactly this. I came to make the same comment, but unsurprisingly someone has beat me to it. In particular, the answer to the "paradox" in a nutshell is this:
> There's no change in belief happening, only two different beliefs. First is "1/2 of the times it is flipped, the coin will be heads". Seconds is "1/3 of the times I am awakened, the coin will be heads".
> There's no change in belief happening, only two different beliefs. First is "1/2 of the times it is flipped, the coin will be heads". Seconds is "1/3 of the times I am awakened, the coin will be heads".