Uncovering class labels based on conditional or prior class probability I dont understand how the lecturer solved this problem. The question is: 

You are working with a dataset that contains descriptions of toxic and
  non-toxic substances. The dataset, which consists of 1000 samples
  from each of the two classes, is described in terms of a class label
  and a number of attributes. The dataset is sorted so that the 1000
  toxic samples come first, followed by the 1000 non-toxic samples.
  Someone tells you that they have confirmed that, for this data set,
  the conditional probability that is gained from knowledge about
  attribute X is not different from the prior class probability.
  Assuming that they are correct, which of the following statements
  could be correct and which could not be correct?
  
  
*
  
*All samples have the same value for attribute X.
  
*All toxic samples have the same value for attribute X while each of the non-toxic samples has its own random value for attribute X.
  

My question is how can either of these answers to this question be correct when the information given is very limited?
The book we use for data mining is Witten and Frank, Data Mining: Practical Machine Learning Tools and Techniques, Morgan Kaufmann, 2011 (3rd ed.).
 A: I would not say that the information is very limited because you have 1000 samples from each group (toxic samples and non-toxic samples).  The conditional probability can be estimated from the data and compared to the prior unconditional probability. It suggests that the covariate is not useful in gaining knowledge about the classes. Now if all samples had the same value for attribute X wouldn't that mean that the data does not tell you anything about attribute X for predicting the class.  The result about the conditional probability could happen for other reasons and (1) need not be true.  But it is possible and if it were true it would explain the result.
Now regarding (2) if all toxic values have a particular value x for the attribute and the nontoxic ones could be different X is useful because whenever X differs from x it would have to be from the nontoxic class. When it is x it could be from either class.But it seems that if X=x you would be best off predicting toxic and you should definitely pick nontoxic when it differs from x.  Being that this is the case the finding that X does not help in prediction of classes could not be true.  So (2) can't happen.  There is an underlying assumption that I am making which is that the sample size is so large that you know that when the sample is toxic it is sure (or very highly probably) that X=x.  Also you have a good estimate of the probability that X is equals x when it is nontoxic.
For answer 3 (not given in the question but added in the comments) a different distribution is described for X but both classes have the same distribution for X so that is also a possible scenario to explain why X is useless for classification.  In (1) both distributions are constant with the same constant while in (3) both distributions are uniform between 1 and 100 half the time and the same constant value the other half of the time.
