At least n Spades in $14$ cards Standard $52$-card deck. $14$ randomly chosen cards.
Total number of Combinations: $C_{52}^{14}$.
The question is: what is the probability of having at least $n$ Spades with the dealt $14$ cards?
When $n=1$, my reasoning is to calculate the $p$ of missing the spades altogether and subtract the result from $1$, so the answer would be:
$$p(S>=1)=1-\frac{C_{39}^{14}}{C_{52}^{14}}$$
When $n=2$, can I use the same method "allowing" just one spade to appear in missing combinations, thus:
$$p(S>=2)=1-\frac{C_{40}^{14}}{C_{52}^{14}}$$
If I use the same logic for "at least 10 spades":
$$p(S>=10)=1-\frac{C_{48}^{14}}{C_{52}^{14}}$$
However, calculations showed me the unbelievably high result $(0.73)$ so I am sensing a mistake somewhere, please check my method and advise.
Thanks to all contributing to my answers, I am still earning my privilege to rate your posts.

Your calculation for $P(S\geq 1)$ is correct.
Your calculation for $P(S\geq 2)$ is incorrect. Your approach of calculating $P(S<2)$ and then taking $P(S\geq 2) = 1-P(S<2)$ is correct, but you did not calculate $P(S<2)$ correctly. $C(40, 14)$ is the number of ways that you can select $14$ cards out of a deck containing, for example, all non-spades and the king of spades. You need to multiply this number by the number of spades you can select.
