Probability ( Introduction to statistical mathematics ) 
There are $5$ red chips and $3$ blue chips in a bowl. The red chips are numbered as $1,2,3,4,5$ and blue chips as $1,2,3$. Two chips are drawn randomly without replacement from the bowl, find the probability that both chips have either the same number or the same colour .

What I tried : 
Only 3 conditions can arise , 


*

*Both are red.

*Both are blue.

*1 red & 1 blue ( chips with same number ) .
Then writing the final probability : 
$\frac{5}{8} \cdot \frac{4}{7}$ (for 2 red's) +  $\frac{3}{8} \cdot \frac{2}{7}$ (for 2 blue's) + $\frac{3}{5} \cdot \frac{1}{3}$ (for 1 red, and 1 blue with same number) .
Is the above solution correct?

Nearly. The last case should be $\frac{6}{8} \cdot \frac{1}{7} = \frac{3}{28}$, as for some reason in that step you forgot that you are still drawing from the whole probability space. You can select any chip besides the ones numbered $4$ or $5$ with your first draw ($\frac{6}{8}$), then based on that chip there is only one corresponding chip of the opposite color in the seven remaining ($\frac{1}{7}$). 
This will change your corresponding final value to be $\frac{2}{7}$. 
