Proving that a Turing machine accepting the string 'bb' is undecidable

Question

Show that the problem of deciding whether an arbitrary Turing machine accepts the string 'bb' is undecidable.

I feel as though this should be decidable even without a turing machine because it is a finite string.

Answer 1

Look into Rice's theorem for a quick and easy proof.

Consider a TM that runs forever on the input aa. How will you know if it ever stops? That is why it is undecidable.