In conditional probability why does conditioning on a specific one yield a very different answer to conditioning on at least one

Question

I was watching the Harvard stat110 course and there was this question

pick a random 2 card hand from a standard deck.

1. what is the probability that it's 2 ace given you have one ace
2. what is the probability that it's 2 ace given you have ace of spade

the result for Q1 is $\frac{1}{33}$ and the result for Q2 is $\frac{1}{17}$

in the lecture he mentioned a hint that it's because one is dealing with at least one while the other is a specific one, but I can't get my head around the intuition behind this idea.

Answer 1

Think of the conditioning as ‘reducing sample space’. Different conditions restrict the sample space differently.

The first case you have $$\{(\text{any } A, \text{anything})\}$$ while the second case you have $$\{(A \text{ of spade},\text{anything})\}$$

Notice that the first one is much larger than the second one while the event spaces are quite similar hence the different probabilities.