In "Computational complexity- A modern approach" book (page 117) for the lemma 7.12 (following) the author mentioned that if the ρ is efficiently computable ρ-coin cannot give probabilistic algorithm a new power. Then he states: "The exercises show that if ρ is not efficiently computable, then a ρ-coin can indeed provide additional power."
Lemma 7.12: A coin with Pr[Heads] = ρ can be simulated by a PTM in expected time O(1) provided the ith bit of ρ is computable in poly(i) time.
The exercise that supposed to show, if ρ is not efficiently computable, then a ρ-coin can indeed provide additional power is the following:
Exercise: Describe a real number ρ such that given a random coin that comes up “Heads” with probability ρ, a Turing machine can decide an undecidable language in polynomial time.
Now, how can we show that if ρ is not efficiently computable, then a ρ-coin can indeed provide an additional power for a probabilistic algorithm to decide some undeniable language in polynomial time?
*I removed the hint because it made me more confused but you can find it at the end of the book.
I strongly believe that this is a research level problem and proper to be discussed in theoretical computer science society because this is a high level question about computational complexity and this question has not been discussed anywhere else.