The Church-Turing Thesis (Hello World tester) contradiction and randomized algorithms
The Church-Turing thesis proves that there is no algorithm (or program)  that says whether a program  written in a language, say , on an input  outputs the sentence Hello World or not. This is his proof, visit Hopcraft and etc.


The Church-Turing thesis is introduced in the book as "the unprovable assumption that any general way to compute will allow us to compute only the partial-recursive functions (or equivalently, what Turing machines or modern-day computers can compute)". It has little to do with undecidability directly. I will treat "The Church-Turing thesis proves that" as a typo that should not exist.




  Please explain where I am wrong about this.


In fact, you are not necessarily wrong at all. It is just that your argument does not lead us anywhere. 

What have been proved in the book is that there is no non-randomized algorithm that can decide whether a non-randomized program on an input outputs "Hello World" or not. No, there is no such adjective "non-randomized" in the original statement in the book. However, it does not matter much whether we add it or not. Why?

Because a randomized algorithm cannot decide anyway. It can only say something like the probability a given program on some given input outputs "Hello world" is less than 23 percent. I will not call that probabilistic prediction a decision. You have not found anything valuable or interesting by considering randomized algorithms. All you have observed is that the proof does not apply to randomized algorithms, which we do not have to consider in the first place anyway.



We might be thinking, wait, there is some information if we can be certain the probability a given program on some given input outputs "Hello world" is less than 23 percent. That could be mean that the undecidability only holds in some relative sense. 

However, the conclusion is still something along the line that there is no algorithm, whether randomized or not, can always provide non-zero information regarding whether a deterministic program on a input outputs "Hello world" or not. You can read the question can a probabilistic Turing machine solve the halting problem? and its answers for more explanations.