Stacking Two Decks

Dec 29, 2012

RAY: Let's say you have two ordinary decks of playing cards, minus the jokers. So, you have a deck of 52 cards and another deck of 52 cards. You take them and you shuffle them up--mix them all up as best you can, one hundred four cards.

And then you divide them into two equal piles. So, you've got a pile of 52 on one side of the table, and a pile of 52 on the other side of the table. You have pile A and pile 2.

What are the chances that the number of red cards in pile A equals the number of black cards in pile 2? That's part one of the question. And then part two of the question: how many cards would you have to look at to be certain of your answer?

RAY: Here’s how you solve it. Let's say by some luck, you shuffled up all these cards and all the red cards wound up in one pile, we'll call that pile A. And for simplicity's sake, we'll call the other pile pile B, and all the black cards wound up in that. Then you would say, well, certainly the number of red cards in deck A, or pile A, equals the number of black cards in pile B. Now, I ask you to construct a scenario where it wouldn't be the case, always.

TOM: How about one and 51?

RAY: Exactly. Take a card out of pile A and donate it to pile B--but when you do that, you must reciprocate. Take a black card from pile B and donate it to pile A, and therefore you have 51 and one, and 51 and one, no matter how you do this.

TOM: And part B of the question: How many cards do you have to look at to verify your answer?

RAY: None! Who's our winner this week?

TOM: Our winner this week is Dominic Matranga from Mobile, Alabama. Congratulations, Dominic! 

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