Probability puzzle on Inference. Couple of months ago, i was asked this weird probability puzzle in an interview.
Problem: There are two persons X and Y. Given a statement, X says it is true w.p $\frac{1}{3}$. and false w.p. $\frac{2}{3}$. Y always agrees with whatever X says.
What is the probability that the statement is actually true? 
A seemingly related question is Truth or Lie?
Please help with a proper explanation. Does the information provided in the question sufficient to infer anything?
 A: As stated, it should be clear that it is unanswerable.
Let $P$ be a statement. Suppose that it is true. What happens? $X$ will tell you it's true with probability $1/3$ and false with probability $2/3$. $Y$ will agree.
Let $P$ be a statement. Suppose that it is false. What happens? $X$ will tell you it's true with probability $1/3$ and false with probability $2/3$. $Y$ will agree.
You're given the same information in both situations. What $X$ tells you is independent of the truth or falsehood of the statement.
Assuming you're not missing any part of the question, I would speculate that the following is what your interviewer was getting at. Sometimes in ML, you will find yourself in a situation where your model has been trained in a way that its predictions become independent of your input data. Think for example about training on a dataset with very imbalanced class ratios-- maybe even only on one label. Would this be detectable? How would one figure out it was happening? Perhaps the interviewer wanted to lead into this sort of discussion.
