# There are 5 red pens, 3 blue pens and 2 green pens in a box. Gary takes at random a pen from the box and gives the pen to his friend. He then takes at random another pen the box. What's the probability that both pens are the same colour?

May 22, 2017

$\frac{14}{45}$

#### Explanation:

$\text{assuming no replacement}$

$\text{total number of pens } = 5 + 3 + 2 = 10$

$P \left(\textcolor{red}{\text{red}}\right) = \frac{5}{10} = \frac{1}{2}$

$P \left(\text{another"color(red)" red}\right) = \frac{4}{9}$

$\Rightarrow P \left(\textcolor{red}{\text{2 reds}}\right) = \frac{1}{2} \times \frac{4}{9} = \frac{2}{9}$

$P \left(\textcolor{b l u e}{\text{blue}}\right) = \frac{3}{10}$

$P \left(\text{another "color(blue)"blue}\right) = \frac{2}{9}$

$\Rightarrow P \left(\textcolor{b l u e}{\text{2 blues}}\right) = \frac{3}{10} \times \frac{2}{9} = \frac{1}{15}$

$P \left(\textcolor{g r e e n}{\text{green}}\right) = \frac{2}{10} = \frac{1}{5}$

$P \left(\text{another "color(green)"green}\right) = \frac{1}{9}$

$P \left(\textcolor{g r e e n}{\text{2 greens}}\right) = \frac{1}{5} \times \frac{1}{9} = \frac{1}{45}$

$\Rightarrow P \left(\textcolor{red}{\text{2 reds")" or " P(color(blue)"2 blues")" or " P(color(green)"2 greens}}\right)$

$= \frac{2}{9} + \frac{1}{15} + \frac{1}{45}$

$= \frac{14}{45}$