Possibility of drawing at least two cards of the same set of 5 cards, from a deck of 40 cards after 20 draws?

Question

I wanted to know the odds of the situation described in the title. I'll explain better: There's a famous card game in which if you have a certain set of 5 cards in your hand, you win the game. My friend was complaining about the fact that he often finds only one of the 5 cards after around 20 draws, and he said that he thinks the videogame is scripted so that you don't draw that specific set of cards. In my opinion, without doing any maths, that's more than possible considering he only played a bunch of matches. So the question is: given that there are only 5 cards of that set in the deck and that you have to draw them all to win the game, and given a deck of 40 cards, what is the formula to calculate the odds of having two of those cards in your hand after 20 draws? Thanks for your answers! :D

Answer 1

If I understand correctly it is $$\frac{C(5,2)C(35,18)}{C(40,20)}\approx 0.3292$$

This is the probability that in the deck of 40 cards, there are 5 "magic" cards (that form a set) and after 20 draws there are exactly 2 "magic" cards.

At least two is $$\frac{C(5,2)C(35,18)}{C(40,20)}+\frac{C(5,3)C(35,17)}{C(40,20)}+\frac{C(5,4)C(35,16)}{C(40,20)}+\frac{C(5,5)C(35,15)}{C(40,20)}\approx 0.8292$$