Before I ask my question I will state the following "simple" problem.
Consider the experiment of drawing cards from a deck. Suppose that we select two cards at random from the desk. When we select the second card, we do not return the first card to the deck. We want to calculate the probability of the second card being spade (event B) given that the first card is spade (event A).
Solution :
Method 1 :
using the definition of conditional probability, the answer is = $\frac{\frac{^{13}C_2}{^{52}C_2}}{\frac{13}{52}} = \frac{4}{17}$
Method 2 :
Here we try to calculate the probability of event B on a reduced sample space (by removing the sample points corresponding to event A (removing the deterministic part from the randomized event) i.e. given that we know the first card selected was a spade, there are now 51 cards left in the deck, 12 of which are spades, thus the required probability = $\frac{12}{51} = \frac{4}{17}$.
Now I have seen that for some problems method 1 is not giving the same answer as method 2 (ex: https://mathoverflow.net/questions/117182/a-problem-on-dtmc). My question is when method 1 will give same answer as method 2 and when not. Hope my confusion is clear from above.