I thought of a problem earlier and I am quite clueless on how to solve it, or begin solving it, because I cannot find a way to easily compute the amount of combinations of $2$ cards that sum to a certain value $x$. Anyway, the first part of the problem is as follows
We have a full deck of $52$ cards, and randomly select $2$ cards from this deck. We look at the cards and compute the sum of the values of the cards, ace being $1$ and K, Q and J being $13$, $12$ and $11$ respectively. We then shuffle the cards back into the deck and randomly select $2$ cards once more. What is the probability that the sum of the value of these $2$ cards, is the same as the sum of the values of the first $2$ selected cards?
This brought me to think of another problem, which is comparable. It is as follows
Let's say we have a full deck of $52$ cards, we randomly select $2$ cards, and we do this twice, yielding $2$ sets of $2$ cards. What is the probability that the sum of the values of the cards in the first set, equals the sum of the values of the cards in the second set?
Again, I'm quite confused about this problem, because I cannot think of an easy way to compute the amount of possible configurations of two cards, that sum to some value $x$.
Any help on solving these problems is appreciated. Furthermore, what would be a good guesstimate for these probabilities that could be given without any computations?