Question

Find the probability for the experiment of drawing two marbles (without replacement) from a bag containing two green, four yellow, and two red marbles. The marbles are of different colors.

Answer 1

The bag contains 2+4+2=8 marbles in total.

We are going to find a probability of the opposite event: drawing marbles of same color.

There are three possibilities:

1. Drawing both green marbles

2. Drawing both yellow marbles

3. Drawing both red marbles

The probability of drawing both green marbles:

$P_1=\frac{2}{8}\cdot \frac{1}{7} = \frac{1}{4}\cdot \frac{1}{7}=\frac{1}{28}$

The probability of drawing both yellow marbles:

$P_2=\frac{4}{8}\cdot \frac{3}{7} = \frac{1}{2}\cdot \frac{3}{7}=\frac{3}{14}$

The probability of drawing both red marbles:

$P_3=\frac{2}{8}\cdot \frac{1}{7} = \frac{1}{4}\cdot \frac{1}{7}=\frac{1}{28}$

So, the probability of drawing marbles of the same color is:

$P=P_1+P_2+P_3=\frac{1}{28}+\frac{3}{14}+\frac{1}{28}=\frac{2}{28}+\frac{3}{14}=\frac{1}{14}+\frac{3}{14}=\frac{4}{14}=\frac{2}{7}$

Now, the probability of drawing marbles of different colors is:

$1-\frac{2}{7}=\frac{5}{7}=0.7142$