Probability of drawing a Queen, without replacement, after a set amount of cards

For an assignment, I have to determine the chance of drawing one queen after a set amount of cards without replacement.

The cards drawn start at 1 and go all the way up to 49 cards:

1 cards = 7.69% chance

2 cards = ____% chance

3 cards = ____% chance

...

49 cards = 100% chance

I have determined that the chance of drawing a queen is 4/52. For every card you draw, the denominator decreases by one. So the probability of drawing one queen by two cards is something along the lines of: 4/52(chance of queen) * 48/51 (chance of drawing something other than queen).

3 draws would be something like: 4/52(chance of queen) * 48/51(chance of no queen) * 47/50(chance of no queen).

The table is drawn out on my paper, and there is a pattern that I am seeing, however the percentage chance of drawing a queen is decreasing after drawing more cards. I can't figure out why. Multiplying by a fraction lower than 1 is lowering my chance of finding a queen.

But wouldn't drawing more cards INCREASE your chances of finding a queen?

I am expecting drawing 48 cards to be something like >95%, but i'm actually getting 0.01% chance. Something is wrong.

I need help understanding this. If anyone could help out, it would be greatly appreciated.

Direct approach. Calculate the probability $Q_n$ of not drawing a queen with n cards and subtract from 1. $Q_n=\frac{48}{52}\frac{47}{51}...\frac{49-n}{53-n}$ and what you want is $P_n=1-Q_n$. Notice that when $n=49$, $Q_n=0$.