I am taking a look at the the following problem: We pick 2 cards at random from a standard deck of cards. Find $P(\text{both aces}| \text{first card is ace})$. I already know that the answer should be computed as follows:
$P(\text{both aces} | \text{first card is ace}) = \dfrac{P(\text{both aces} \cap \text{first card is ace})}{P(\text{first card is ace})} = \dfrac{P(\text{both aces})}{P(\text{first card is ace})} = \dfrac{\binom{4}{2}/\binom{52}{2}}{1-\binom{48}{2}/\binom{52}{2}} = \dfrac{1}{33}$
But my intuition tells me that if I already picked one ace, there are just 3 aces left in the remaining 51 cards which leads to a probability of $\dfrac{1}{17}$. Why is this reasoning wrong?