# 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?

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Jim G. Share
May 22, 2017

$\frac{14}{45}$

#### Explanation:

$\text{assuming no replacement}$

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

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

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

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

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

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

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

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

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

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

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

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

$= \frac{14}{45}$

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