Chance with cards.

Hi there. I really hope I'm posting the right place and right way since this is my first time here, please excuse me if this has been done wrong and tell me how to fix, but moving on.

I'm a student who while playing a card game with 3 friends got curious about how the math with probability works in this particular game. In the game each player has 13 cards and one has to choose an ace, the player with the chosen ace in hand will end up playing with the one choosing for the rest of the game. In the game you make a stack of 3 cards which the player choosing the ace can exchange for their own. If the player choosing the ace chooses an ace which lies in the stack of cards they exchange with, they will end up playing the game alone against 3 instead, however, this is incredibly rare, I just want to find out how rare.

I started by trying to find the following:
The possibility of one of the four aces from the deck of the 55 cards (it has 3 jokers) ending up in the stack of the three randomly chosen cards, and after that I wanted to find out what the chances were of the player choosing exactly that ace after it had ended up there. To do the first part I thought I could do the following:
3/55*4/55, because there are 3 cards in the stack and 4 aces in the 55 cards, but I was then told that it seems unlikely that this is the answer since that after you draw the first card there are only 54 remaining and so on. Because of that I thought I could it like this: (1/(4*55)*(1/(4*54)*(1/(4*53) but that doesen't really seem to make sense after thinking about it, and looking at the result. And so after browsing various math sites I figured I would try asking on a forum instead.

So, in conclusion, I want to find out the possibility of one of the 4 aces ending up in the stack of 3 randomly chosen cards, and that ace then being chosen by the player who's choosing one of the 4. Thanks in advance.

The description probably seems a bit lackluster due to the fact that I was hesitant to make an in depth explanation of the game since I thought it to be irrelevant, because of that I sort of made one but dind't go in depth so yeah... But basically: After every player has received 13 cards a stack of the 3 remaining are made. The player choosing the ace may, after choosing an ace, swap these 3 cards for 3 of their own, this makes it possible that the player chooses an ace that is in the stack, and how the chance of this happening is found is the reason I'm posting here. And to answer your questions:

Question:
what if that player already has the 4 aces?
That is then an irrelevant scenario not important to my "case". What I want to find out is what the chances are of the ace ending up in the stack and later being the ace chosen among the 4 that are possible to select.

In the game you make a stack of 3 cards which the player
choosing the ace can exchange for their own.

Questions:
WHEN is this stack made? HOW is it made?
Do you mean "exchange for HIS own"?

After the cards are given to the players, with the remaining cards. Yes, with his own, sorry for explaining that poorly.

If the player choosing the ace chooses an ace which lies
in the stack of cards they exchange with, they will end up
playing the game alone against 3 instead

Question:
what does this have to do with probability calculation?

I may have misunderstood the whole thing, but, since there is a statistical chance of the previously mentioned event happening, you should be able to calculate exactly what the chances of this happening is, and I was hoping to receive help in order to figure out how it's calculated, the bad explanation was in order to give an idea of the game and what I wanted to calculate.