In a set of 5 bottles,1 has a fracture.If you select a pair of bottles, probability that the fractured bottle is chosen is? Its also mentioned that this question is an example of Sampling without replacement
My question is , by the general method of how I do these kind of problems , I would assume that there are two possibilities


*

*You chose a defective bottle and then  a non defective which will have the probability of 
  1/5 X 4/4 = 4/20

    OR


*Choosing a non defective one first and then a defective one which will have the probability of
4/5 x 1/4 =4/20

Total probability is 8/20= 2/5 which doesn't make sense at all, since there is just one bottle to begin with
But the problem is I cant find a fault in my logic. Though I feel both the options are redundant , isn't both two different ways of selecting the pair?. Or is the fact that the phrase "one after the other" is absent a reason on why this might be a wrong approach?
    Thank you in advance


Your logic is perfectly sound. The answer is $\frac{2}{5}$. I think you find it odd that this is more than the probability of choosing one bottle and it being fractured, but this is actually what you must expect. 
Think of the bottles trying to 'clear an exam'. If the exam is easy, more students tend to pass. Thus, each student has a higher chance of passing. You allowing more bottles to 'pass' (choosing more bottles) gives more chance for the fractured bottle to pass too! In the extreme case where all bottles pass, this becomes obvious.
