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:

`\frac{\dbinom20\dbinom30\dbinom42}{\dbinom{2+3+4}2}=\frac6{\dbinom92}=\frac16`

1 white, 1 black:

`(\dbinom21\dbinom30\dbinom41)/(\dbinom92)=(2*4)/(\dbinom92)=\frac29`

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):

`(\dbinom22\dbinom30\dbinom40+\dbinom21\dbinom31\dbinom40)/(\dbinom92)=(1+2*3)/(\dbinom92)=\frac7{36}`