Four cards are randomly chosen without replacement from an ordinary deck of 52 playing cards.

Question

Four cards are randomly chosen without replacement from an ordinary deck of 52 playing cards. What is the chance of all chosen cards not being aces?

I found a similar problem here but the provided answer doesn't help me. So, all the card that aren't aces are 48. But I don't know if all cases are 52 or $^{52}C_4$. Therefore, I am wondering if the probability is $P(A)=\frac{48}{52}$ or $P(A)=\frac{48}{^{52}C_4}$. The second probability should be the right one but it looks too low to be realistic. Which is the right answer?

Answer 1

There are $52$ cards in total, but you can effectively only pick from $48$ of them (the non-aces).

This gives the probability as:

$$\frac{\binom{48}{4}}{\binom{52}{4}}$$