Q:

Probability, why my solution doesn't work out? (P of drawing a pair)

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. 

A:

The probability that the first card is a 2, is 4/52, not 1/52 (there are four cards with a "value" of 2 and 52 cards total). Your reasoning works out, with this correction. 

