[Math] what is the conditional probability that the card following it is the ace of spades

Question

i have seen a couple of different answers for this question, but i still cant understand why.
Suppose that an ordinary deck of 52 cards is shuffled and the cards are then turned over one at a time until the first ace appears. Given that the first ace is the 20th card to appear, what is the conditional probability that the card following it is the ace of spades?
this is what i was trying.
p(following is the ace of spades| first ace is the 20th card)=p(following is the ace of spades ∩ first ace is the 20th card)/p(first ace is the 20th card)=(4C1/32C1*3C1/31C1)/(4C1/32C1)=((4!/3!)/(32!/31!)*(3!/2!)/(31!/30!))/((4!/3!)/(32!/31!))

Answer 1

There's a $\frac 14$ probability that the first ace is $A\spadesuit$, in which case the answer would be $0$. Otherwise, there are $32$ remaining cards of which one is $A\spadesuit$ so $\frac 1{32}$ in that case. Hence $$\frac 14\times0+\frac 34\times \frac 1{32}=\frac 3{128}$$