Question

A box contains 2 white, 3 red, and 4 black marbles. you choose two marbles and each are chosen at random. find the probability that:
1. both marbles are black?
2. one marble is white and the other is black?
3. one marble is white and the other is not black?

Answer 1

2 black marbles:
$\dfrac{\dbinom20\dbinom30\dbinom42}{\dbinom{2+3+4}2}=\dfrac6{\dbinom92}=\dfrac16$

1 white, 1 black:
$\dfrac{\dbinom21\dbinom30\dbinom41}{\dbinom92}=\dfrac{2\times4}{\dbinom92}=\dfrac29$

1 white, 1 non-black (this means you draw either 2 white, or 1 white and 1 red; we add the events' probabilities because they are disjoint events):
$\dfrac{\dbinom22\dbinom30\dbinom40+\dbinom21\dbinom31\dbinom40}{\dbinom92}=\dfrac{1+2\times3}{\dbinom92}=\dfrac7{36}$