TITLE: Probability that at least one of 2 balls taken randomly from a pile of 2 red and 3 black is red
QUESTION [8 upvotes]: I get 2 different answers, depending how I approach this, and I need help to see why the error arises.
One solution is to calculate unfavorable combinations probability and substract from 1:
$$1-\frac{C_3^2}{C_5^2}=\frac{7}{10}$$
The other solution is when calculating favorable combinations, to first choose one of the 2 reds, and then choose one of the remaining 4:
$$\frac{C_2^1 \times C_4^1}{C_5^2} = \frac{8}{10}$$
Which is obviously $\neq \frac{7}{10}$.

REPLY [4 votes]: For what it's worth, 7/10 (aka 14/20, reduced) is correct through another method:

I would've made this a comment since it doesn't truly answer your question, but comments can't have pretty pictures.