Probability that the number of different-colored cards in two decks is equal After googling a bit and filling up a couple pieces of scratch paper, I haven't found a way to prove my intuition of the solution to this problem.
Take two normal identical decks of cards (52 each), combine them, and shuffle them randomly. Then cut the combined deck of 104 cards in half, giving you new decks A & B.
1) What is the probability that the number of red cards in deck A is equal the number of black cards in deck B?
2) How many cards would one have to check to ensure that those two numbers are equal?
Say we call X: the number of red cards in deck A, X': the number of black cards in deck A, Y: the number of red cards in deck B, and Y': the number of black cards in deck B.
For part 1, my intuition says that P(X == Y') = 1, since X is always equal to Y':
X = 0    X' = 52   Y = 52   Y' = 0,
X = 1    X' = 51   Y = 51   Y' = 1,
...,
X = 51   X' = 1    Y = 1    Y' = 51,
X = 52   X' = 0    Y = 0    Y' = 52.

And for part 2, my intuition says 52, because you couldn't know if you've seen all the cards of whichever color you're looking for in whichever deck you're looking through until you look at all the cards in that deck.
So my ultimate question is twofold: is my intuition correct, and how would I prove those answers?
Does proving that P(X == Y') = 1 have something with a cdf and/or stats-related calculations, or could you just use an induction-ish (sort of like the above table) proof?
 A: Your answer, for the first part is correct. In order to prove it, you just need to pay attention to the hint, given by "awkward". If you write the statement, for which you want to find the probability, you would see that it always happens, no matter how you shuffle the cards.
For the second part, however, your answer is not correct. When you know an event definitely happens, then you do not need to gain information to prove it. If you check cards, it means that you are gaining information about an outcome, which is not for sure.
A: Your intuition about part 1 is correct: it will always be the case that the number of red cards in deck A is equal the number of black cards in deck B.
Only a counting argument is needed to show this. You could summarise the argument that's implicit in your table of values as follows:


*

*If there are $x$ red cards in deck A, then this deck must contain $52 - x$ black cards. 

*Since there were $52$ black cards to begin with, the remaining $x$ black cards will be in deck B.

*Therefore there are $x$ red cards in deck A, and $x$ black cards in deck B.


As for the second part, you don't need to check any cards at all to know that this is the case. We have shown that no matter how the cards were shuffled the statement will be true.
