deck of 52 cards: split into piles, and compare numbers of

[stars584]

1) When you have a deck of 52 cards and you divide it into 2 equal piles (26 cards each), why is there the same number of red cards in pile one as black cards in pile two?

Would this be because the deck is even number? Because the same card has 2 red cards and two black cards...?

2) Divide the cards into piles by color. Take any number of cards from pile 1 (the red pile) and place in pile 2 (the black pile), and vice versa. Why would you have the same number of black cards in pile 1 as red cards in pile 2?

The probablity of getting the same number of cards is the same. Can I use n! or 1/n!-1? to determine this?

 

[daon]

For 1:

Split the cards up into two piles, Pile 1 and Pile 2. Say Pile 1 has x red cards and y black cards. Since there are 26 red cards and 26 black cards, Pile 2 must have 26-x red cards and 26-y black cards. Now, Pile 1 also has 26 cards, so 26=x+y.

We already know that Pile2 has 26-x red cards. But Pile 1 has y black cards, and solving for y above we get y=26-x, so they are equal. Similarly, Pile 1 has x red cards, and Pile 2 has 26-y or x black cards.

For 2: This is just a special case of number 1, assuming that the piles always have the same number of cards.