The task is simple, the probability of drawing a pair of cards. You draw two cards from a stack, what is the chance that you get two kings or two fours.
My idea was the following. There are 13 different valued cards. The probability of getting lets say a pair of two is following.
$\frac{1}{52} \cdot \frac{3}{51}$
The chance of getting first card 2 is $\frac{1}{52}$, the chance of getting one of the three other cards that would make this a pair is $\frac{3}{51}$. That sounds reasonable to me. And now to account for all 13 different types you just multiply this with 13. Or add it up 13 times.
$13(\frac{1}{52} \cdot \frac{3}{51})$
This is wrong and I don't understand why. The probability of getting a random pair should be the sum of getting every type of pair.
A correct solution would be the following.
$\frac{52 \cdot 3}{52 \cdot 51}$
I try to stick with the 'count the number of beneficial outcomes in every step and multiply method' but I got stuck on why my way of thinking didn't work out here.